#7507461

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,26,29,24)(20,21,32,23)(27,30,31,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,25,26)(20,31,32,27)(21,30,23,28);(1,4,5,6)(2,3,7,8)(10,14)(13,15)(17,20)(18,23)(19,26)(21,25)(22,32)(24,29)(27,30)(28,31);(2,7)(3,8)(11,16)(13,15)(17,20)(18,21)(19,27)(22,32)(23,25)(24,28)(26,30)(29,31);(17,21,22,23)(18,20,25,32)(19,28,29,30)(24,27,26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7414657);