#378781

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

820 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,33,42)(15,20,17,22,26,35,32,41)(16,21,30,39,31,40,29,38)(25,34,28,37,43,44,27,36);(6,9)(10,12)(13,15,18,17,19,26,33,32)(14,20,23,22,24,35,42,41)(16,27,29,43,31,28,30,25)(21,36,38,44,40,37,39,34);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,28)(18,30)(19,31)(20,34)(22,37)(23,39)(24,40)(26,43)(27,32)(29,33)(35,44)(36,41)(38,42);(6,9)(10,12)(13,17,18,26,19,32,33,15)(14,22,23,35,24,41,42,20)(16,28,30,43,31,27,29,25)(21,37,39,44,40,36,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285977);