#7492901

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324532
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	#59818	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400097);