#7513042

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=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, 7420238);