#7621305

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 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)(19,27,24,21)(20,22,26,31)(28,30,32,29);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,24)(18,21,25,27)(20,28,26,32)(22,29,31,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,22)(19,28)(21,29)(23,26)(24,32)(25,31)(27,30);(10,13)(14,16)(17,21,31,32,23,27,22,28)(18,24,20,30,25,19,26,29);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,20,25,26)(19,29,24,30)(21,28,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=ghi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7528501);