#483996

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
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

429 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 158 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)(4,7,12,15,13,11,16,14)(17,18,20,27,22,28,34,25)(19,29,32,44,33,43,30,24)(21,36,35,26,39,23,37,41)(31,42,48,46,38,45,47,40);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16)(18,23)(20,34)(21,37)(24,40)(25,41)(26,27)(28,36)(29,45)(30,32)(31,47)(35,39)(38,48)(42,43)(44,46);(1,4,10,16,6,13,5,12)(2,7,8,14,3,11,9,15)(17,19)(18,24,28,44)(20,32)(21,38)(22,33)(23,40,36,46)(25,43,27,29)(26,45,41,42)(30,34)(31,39)(35,47)(37,48);(2,3)(7,11)(8,9)(14,15)(17,20,22,34)(18,25,28,27)(19,30,33,32)(21,35,39,37)(23,26,36,41)(24,29,44,43)(31,47,38,48)(40,42,46,45)

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=cgh,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	#72286048, #72286070, #72286247, …
maximal normal subgroup of	#72286048, #72286070, #72286247, …
maximal quotient of	#72343445, #72343473, #72343500, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391192);