#6516848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3427 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=h,e^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303498, #72303537, #72303567
maximal normal subgroup of	#72303498, #72303537, #72303567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424044);