#366921

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1160 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)(15,23)(16,28)(17,21)(18,31)(19,26)(20,30)(22,24)(25,40)(27,35)(29,39)(32,38)(33,37)(34,43)(36,42)(41,44);(6,9)(10,11)(13,15,24,27,17,26,18,32)(14,19,22,38,21,23,31,35)(16,25,33,42,29,41,30,43)(20,36,28,44,37,34,39,40);(2,4)(13,16,17,29)(14,20,21,37)(15,25,26,41)(18,33,24,30)(19,34,23,36)(22,39,31,28)(27,43,32,42)(35,40,38,44);(5,7,8,12)(6,10,9,11)(14,21)(19,23)(20,37)(22,31)(28,39)(34,36)(35,38)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274117);