#6409514

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1333 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)(10,14)(11,12)(17,18,23,27,22,26,30,32)(19,24,28,25,21,31,20,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,30)(21,22)(23,28)(25,27)(26,31)(29,32);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,23)(21,30)(22,28)(24,32)(26,29)(27,31);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,21,22,19)(18,24,26,31)(20,30,28,23)(25,32,29,27);(9,12,13,11)(10,15,14,16)(19,21)(20,28)(24,31)(25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6316710);