#6412821

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(17,18,23,27,22,26,32,29)(19,25,28,20,21,24,31,30);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,21)(18,24,26,25)(20,27,30,29)(23,28,32,31);(10,14)(11,15)(17,20,23,24,22,30,32,25)(18,19,27,28,26,21,29,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,22,19)(18,25,26,24)(20,29,30,27)(23,31,32,28);(2,5)(6,8)(18,26)(19,21)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=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, 6320017);