#356160

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263356);