#484144

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
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	#2218	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

402 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216726, #72216740, #72216779, …
maximal normal subgroup of	#72216726, #72216740, #72216779, …
maximal quotient of	C₂×#484144, #72318588, #72318765, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391340);