#7596134

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59213	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,31)(21,26)(22,27)(23,28)(25,29)(30,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,27)(26,31)(28,32);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,30,23,29,21,19)(18,24,26,25,28,32,27,31);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,19,21,29,23,30,22,20)(18,25,27,24,28,31,26,32);(2,8)(6,7)(17,21,23,22)(18,26,28,27)(19,20,30,29)(24,31,32,25)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282784, #72282893, #72284227
maximal normal subgroup of	#72282784, #72282893, #72284227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7503330);