#7622838

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622838
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3864 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)(10,23)(12,26)(13,25)(15,18)(16,29)(21,30)(22,31)(24,32)(27,28);(1,2)(3,8)(4,15)(5,18)(6,9)(7,14)(10,17)(11,21)(12,27)(13,28)(16,23)(19,29)(20,26)(22,25)(24,31)(30,32);(4,16)(7,19)(12,24)(14,29)(15,23)(20,30)(26,32)(27,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,20)(9,21)(10,22)(14,26)(15,27)(16,24)(17,25)(18,28)(19,30)(23,31)(29,32);(3,11)(8,21)(12,24)(13,25)(20,30)(22,28)(26,32)(27,31)

Polycyclic presentation:

2^9|f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,f^b=fh,g^b=gi,h^b=hi,f^c=fi,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	#72226794

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530034);