#414909

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

996 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322105);