#7065143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972339);