#7062572

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(13,14,19,23,18,24,15,20)(16,27,25,21,30,35,28,38)(17,31,32,36,34,22,26,39)(29,44,41,40,33,43,42,37);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,24,23)(16,28,30,25)(17,32,34,26)(21,27,38,35)(22,39,31,36)(29,41,33,42)(37,44,40,43);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,33)(18,30)(19,28)(20,35)(22,40)(23,27)(24,38)(26,42)(29,34)(31,37)(32,41)(36,43)(39,44);(2,4)(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,34)(19,32)(20,36)(21,37)(23,39)(24,31)(25,41)(27,43)(28,42)(30,33)(35,44)(38,40);(6,10)(9,11)(13,15,18,19)(14,23,24,20)(16,25,30,28)(17,32,34,26)(21,27,38,35)(22,36,31,39)(29,42,33,41)(37,44,40,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6969768);