#7623153

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623152
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215809, #72215833, #72215848
maximal normal subgroup of	#72215809, #72215833, #72215848
maximal quotient of	#72226791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530349);