#6782891

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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)(19,26)(20,24)(21,31)(22,25)(23,29)(27,28)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,31)(21,24)(22,25)(23,30)(26,28)(29,32);(10,13)(14,16)(17,19,20,28,22,30,21,29)(18,23,31,32,25,27,24,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,31)(21,24)(22,25)(26,29)(27,30)(28,32);(3,8)(5,7)(10,13)(14,16)(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32)

Polycyclic presentation:

2^9|c^2=gi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6690087);