#6971205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907889
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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,d^c=dfhi

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, 6878401);