#189750

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96946);