#6456012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#26619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363208);