#405188

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312384);