#392508

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,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, 299704);