#7623063

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623063
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60461	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215833
maximal normal subgroup of	#72215833
maximal quotient of	#72226797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530259);