#6810579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#3219	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(7,11)(8,10)(9,12)(13,14,19,25)(15,20,28,37)(16,22,31,41)(17,24,18,23)(21,33,39,30)(26,35,29,40)(27,42,32,36)(34,43,38,44);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,32)(18,27)(19,28)(21,38)(22,40)(23,42)(24,36)(25,37)(26,31)(30,43)(33,44)(34,39)(35,41);(1,2,4,3)(7,12)(9,11)(13,16)(14,21)(15,26)(17,30)(18,33)(19,31)(20,34)(22,23)(24,41)(25,39)(27,43)(28,29)(32,44)(35,42)(36,40)(37,38);(6,10)(9,11)(13,16)(14,22)(15,26)(17,33)(18,30)(19,31)(20,35)(21,23)(24,39)(25,41)(27,44)(28,29)(32,43)(34,42)(36,38)(37,40);(1,3,4,2)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,36,37,42)(21,22,39,41)(26,43,29,44)(34,40,38,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,e^2=fh,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, 6717775);