#7610220

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59099	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(20,28)(23,32)(26,30)(29,31);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,25)(19,22,27,24)(20,29,28,31)(23,30,32,26);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,27)(18,22,25,24)(20,30,28,26)(23,29,32,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,27,30,21,28,19,26)(18,23,22,31,25,32,24,29);(2,6)(7,8)(17,19,21,27)(18,22,25,24)(20,26,28,30)(23,31,32,29)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256835, #72256858, #72256968
maximal normal subgroup of	#72256835, #72256858, #72256968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7517416);