#378926

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

860 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,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, 286122);