#6850713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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	#24129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6757909);