#6850697

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682500
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(17,18,21,25)(19,22,27,24)(20,28,30,31)(23,29,32,26);(1,2)(3,6)(4,5)(7,8)(11,12)(14,15)(17,19,21,27)(18,22,25,24)(20,26,30,29)(23,31,32,28);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,19,26,21,30,27,29)(18,23,24,28,25,32,22,31);(1,3,4,7)(2,6,5,8);(9,12,13,11)(10,15,16,14)(17,19,21,27)(18,22,25,24)(20,29,30,26)(23,28,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6757893);