#7513798

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,31)(23,25)(26,27)(28,32)(29,30);(1,3,6,2)(4,7,5,8)(17,19)(18,24)(20,30)(21,27)(22,29)(23,28)(25,32)(26,31);(10,13)(14,16)(17,20,22,18)(19,26,29,32)(21,31,23,25)(24,28,30,27);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,21,22,23)(18,25,20,31)(19,27,29,28)(24,32,30,26);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,20,31)(19,28,29,27)(24,26,30,32)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

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, 7420994);