#6686971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6594167);