#7621174

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,26,29,21)(20,31,24,22)(27,30,32,28);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,21,25,26)(20,32,24,27)(22,28,31,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,24)(18,22,25,31)(19,27,29,32)(21,30,26,28);(10,13)(14,16)(17,21,22,27,23,26,31,32)(18,19,20,30,25,29,24,28);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,24,25,20)(19,28,29,30)(21,27,26,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=ghi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7528370);