#6783207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1615 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)(3,7)(4,6)(5,8)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,28)(21,26)(23,25)(24,31)(27,30)(29,32);(10,13)(14,16)(17,20,22,29,23,30,21,31)(18,24,25,28,26,32,19,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,31)(28,29)(30,32);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,32,28)

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=i,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6690403);