#6677146

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(7,10)(8,12)(9,11)(13,14)(15,25)(16,29)(17,24)(18,20)(19,23)(21,28)(22,30)(26,38)(27,41)(31,36)(32,37)(33,34)(35,42)(39,40)(43,44);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(14,20)(15,26)(16,30)(23,24)(28,33)(31,32)(35,44)(39,41);(2,4)(13,15,18,26)(14,21,20,34)(16,27,30,40)(17,28,19,33)(22,35,29,44)(23,25,24,38)(31,42,32,43)(36,41,37,39);(5,8,10,9)(6,11,7,12)(13,16)(14,22)(15,27)(17,32)(18,30)(19,31)(20,29)(21,35)(23,37)(24,36)(25,39)(26,40)(28,43)(33,42)(34,44)(38,41);(2,4)(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,23,20,24)(15,28,26,33)(16,31,30,32)(21,25,34,38)(22,36,29,37)(27,42,40,43)(35,41,44,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6584342);