#393580

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 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)(13,14,18,25,20,26,19,24)(15,21,29,38,31,39,30,37)(16,23,17,22,34,41,33,42)(27,43,44,35,32,36,28,40);(2,4)(13,15,18,29,20,31,19,30)(14,21,25,38,26,39,24,37)(16,32,17,28,34,27,33,44)(22,40,41,43,42,35,23,36);(6,9)(10,12)(13,16,19,33,20,34,18,17)(14,22,25,41,26,42,24,23)(15,27,30,28,31,32,29,44)(21,35,38,36,39,40,37,43);(5,7,8,11)(6,10,9,12)(13,17,18,34,20,33,19,16)(14,23,24,42,26,41,25,22)(15,28,29,27,31,44,30,32)(21,36,37,35,39,43,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,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, 300776);