#393652

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 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)(13,14,19,25,20,24,18,26)(15,21,30,37,29,38,31,36)(16,23,32,39,33,40,17,22)(27,35,28,34,41,43,42,44);(1,2,3,4)(13,15,18,30,20,29,19,31)(14,21,25,36,24,38,26,37)(16,27,17,28,33,41,32,42)(22,34,23,35,39,44,40,43);(6,9)(10,12)(13,16,18,17,20,33,19,32)(14,22,25,23,24,39,26,40)(15,27,30,28,29,41,31,42)(21,34,36,35,38,44,37,43);(5,7,8,11)(6,10,9,12)(13,17,18,33,20,32,19,16)(14,23,25,39,24,40,26,22)(15,28,30,41,29,42,31,27)(21,35,36,44,38,43,37,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=gi,d^2=f,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,e^b=ei,h^b=hi,d^c=di

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, 300848);