#7536250

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,32)(20,27,21,29)(24,30,25,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,32,31);(10,14)(13,15)(17,20)(18,24)(21,22)(25,26)(27,29)(30,31);(2,5)(6,8)(10,14)(13,15)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285106, #72285539
maximal normal subgroup of	#72285106, #72285539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7443446);