#7526269

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,28)(21,30)(22,26)(24,29)(25,31)(27,32);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,22,20)(18,23,26,28)(21,27,31,29)(24,30,32,25);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,20,22,19)(18,24,26,32)(21,27,31,29)(23,30,28,25);(2,8)(6,7)(19,20)(23,28)(24,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256848, #72256888, #72256985
maximal normal subgroup of	#72256848, #72256888, #72256985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7433465);