#7556672

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1604 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)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,29)(31,32);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,23,25,27)(20,28,26,32)(22,29,24,31);(10,14)(15,16)(17,20,21,26)(18,24,25,22)(19,28,30,32)(23,31,27,29);(9,11,13,12)(10,15,14,16)(18,25)(19,29)(22,24)(23,28)(27,32)(30,31);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(18,25)(19,30)(20,26)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7463868);