#7620794

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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)(4,6)(5,8)(10,14)(11,12)(17,18,23,26)(19,21,25,24)(20,22,27,32)(28,30,31,29);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,23,25)(18,24,26,21)(20,28,27,31)(22,29,32,30);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,28)(21,30)(23,27)(24,29)(25,31)(26,32);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,21,22,31,23,24,32,28)(18,25,27,29,26,19,20,30);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(17,22,23,32)(18,20,26,27)(19,29,25,30)(21,31,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fghi,e^2=gi,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527990);