#393796

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,d^b=di

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, 300992);