#7561921

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469117);