#7541610

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7448806);