#6810186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,18,31)(16,26,28,33)(17,22,30,38)(21,32,36,24)(25,34,29,37)(27,40,42,43)(35,41,44,39);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,25,30,29)(21,33,36,26)(22,34,38,37)(27,39,42,41)(35,43,44,40);(1,2,4,3)(7,12)(9,11)(13,16)(14,21)(15,24)(17,27)(18,32)(19,28)(20,33)(22,35)(23,36)(25,39)(26,31)(29,41)(30,42)(34,43)(37,40)(38,44);(6,10)(9,11)(13,17,19,30)(14,22,23,38)(15,25,18,29)(16,27,28,42)(20,34,31,37)(21,35,36,44)(24,39,32,41)(26,40,33,43);(1,3,4,2)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,29,30,25)(21,33,36,26)(22,37,38,34)(27,41,42,39)(35,40,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717382);