#7513041

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

1324 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)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31);(1,2)(3,6)(4,8)(5,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,22)(19,25)(23,27)(24,29)(26,31)(30,32);(2,7)(6,8)(10,13)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(20,28)(24,29)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,e^2=fgi,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, 7420237);