#6459308

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6366504);