#7062378

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969574);