#6807166

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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(13,14,17,19)(15,23,24,36)(16,27,28,21)(18,31,30,22)(20,32,33,26)(25,39,38,37)(29,35,43,42)(34,40,41,44);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,24)(16,28)(22,31)(26,32)(29,43)(34,41)(37,39);(1,2,4,3)(13,15)(14,20)(16,25)(17,24)(18,32)(19,33)(21,34)(22,36)(23,31)(26,30)(27,41)(28,38)(29,44)(35,39)(37,42)(40,43);(6,10)(9,11)(13,16,17,28)(14,21,19,27)(15,25,24,38)(18,29,30,43)(20,34,33,41)(22,35,31,42)(23,37,36,39)(26,40,32,44);(1,3,4,2)(7,12)(9,11)(15,24)(16,28)(20,33)(21,27)(23,36)(26,32)(29,43)(35,42)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6714362);