#6670554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6577750);