#6516846

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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3525 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=h,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	#72303149, #72303332, #72303567
maximal normal subgroup of	#72303149, #72303332, #72303567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424042);