#416893

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

1058 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 324089);