#6371009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=f,b^a=bf,c^a=cg,g^a=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311200, #72311478, #72311482
maximal normal subgroup of	#72311200, #72311478, #72311482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278205);