#392261

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312288, #72313113, #72313118
maximal normal subgroup of	#72312288, #72313113, #72313118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299457);