#415160

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

882 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)(7,11)(8,9)(10,12)(13,14,19,23,18,22,17,30)(15,24,31,34,27,38,26,20)(16,21,28,41,29,35,32,36)(25,37,39,33,40,44,42,43);(1,2)(3,4)(13,15,17,26,18,27,19,31)(14,20,23,24,22,34,30,38)(16,25,28,39,29,40,32,42)(21,33,36,37,35,43,41,44);(2,4)(6,9)(10,11)(13,16,17,28,18,29,19,32)(14,21,23,36,22,35,30,41)(15,25,26,39,27,40,31,42)(20,33,24,37,34,43,38,44);(5,7,8,12)(6,10,9,11)(14,22)(20,34)(21,35)(23,30)(24,38)(33,43)(36,41)(37,44)

Polycyclic presentation:

2^9|a^2=efh,b^2=e,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 322356);