#414998

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322194);