#483965

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33748	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18,20,27,22,28,34,25)(19,29,32,24,33,42,30,44)(21,36,35,26,39,23,37,41)(31,45,47,40,38,43,46,48);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(18,23)(20,34)(21,37)(24,40)(25,41)(26,27)(28,36)(29,43)(30,32)(31,46)(35,39)(38,47)(42,45)(44,48);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(17,19)(18,24,28,44)(20,32)(21,38)(22,33)(23,40,36,48)(25,29,27,42)(26,45,41,43)(30,34)(31,39)(35,46)(37,47);(2,3)(7,11)(8,9)(15,16)(17,20,22,34)(18,25,28,27)(19,30,33,32)(21,35,39,37)(23,26,36,41)(24,42,44,29)(31,46,38,47)(40,43,48,45)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216730, #72216810, #72216968, …
maximal normal subgroup of	#72216730, #72216810, #72216968, …
maximal quotient of	C₂×#483965, #72318467, #72318495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391161);