#6401693

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#2671	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21,22,25,20,30)(19,26,27,28,29,24,31,32);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,24,25,26)(20,31,23,27)(21,32,30,28);(10,14)(13,16)(17,19)(18,24)(20,31)(21,32)(22,29)(23,27)(25,26)(28,30);(1,3,4,7)(2,6,5,8)(17,20,22,23)(18,21,25,30)(19,27,29,31)(24,28,26,32);(2,5)(6,8)(17,21,23,25,22,30,20,18)(19,28,27,24,29,32,31,26)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308889);