#6989253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884873
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	#36360	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896449);