#392423

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1366 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)(7,9)(8,10)(11,12)(13,14,19,23,18,21,17,33)(15,24,28,35,25,38,27,20)(16,29,32,36,31,43,30,22)(26,40,42,44,39,34,41,37);(5,7,8,11)(6,9,10,12)(13,15,19,28,18,25,17,27)(14,20,33,38,21,35,23,24)(16,26,32,42,31,39,30,41)(22,34,43,44,36,40,29,37);(6,10)(9,12)(14,21)(15,25)(22,36)(23,33)(26,39)(27,28)(29,43)(41,42);(2,4)(13,16,17,30,18,31,19,32)(14,22,23,29,21,36,33,43)(15,26,27,41,25,39,28,42)(20,34,24,37,35,40,38,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299619);