#378927

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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)(13,14,19,25,20,26,18,24)(15,21,17,33,29,36,28,23)(16,22,31,39,32,40,34,41)(27,35,30,37,42,44,43,38);(5,7,8,12)(6,9,10,11)(13,15,18,28,20,29,19,17)(14,21,24,23,26,36,25,33)(16,30,31,42,32,43,34,27)(22,37,39,44,40,38,41,35);(1,2,3,4)(13,16)(14,22)(15,27)(17,30)(18,34)(19,31)(20,32)(21,35)(23,38)(24,41)(25,39)(26,40)(28,43)(29,42)(33,37)(36,44);(6,10)(9,11)(13,17,19,29,20,28,18,15)(14,23,25,21,26,33,24,36)(16,30,31,42,32,43,34,27)(22,38,39,35,40,37,41,44)

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=dei,c^b=cg,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, 286123);