#6668598

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2721 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)(13,14,18,23)(15,19,25,32)(16,21,27,35)(17,20,29,22)(24,31,38,43)(26,30,39,33)(28,34,40,36)(37,41,44,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,30)(21,31)(22,33)(23,32)(27,38)(28,37)(29,39)(34,41)(35,43)(36,42)(40,44);(2,4)(14,20)(17,29)(19,30)(21,34)(22,23)(26,39)(28,40)(31,41)(32,33)(35,36)(37,44)(42,43);(6,10)(9,11)(13,16,18,27)(14,21,23,35)(15,24,25,38)(17,28,29,40)(19,31,32,43)(20,34,22,36)(26,37,39,44)(30,41,33,42);(2,4)(7,12)(9,11)(15,25)(16,27)(19,32)(21,35)(26,39)(28,40)(30,33)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6575794);