#6684875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592071);