#392256

order	512
nilpotency class	3
isoclinism	isoclinic class of #184979
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,f^c=fi,h^d=hi

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, 299452);