#6373541

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2146 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280737);