#6403153

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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)(3,5)(4,6)(7,8)(9,10,14,12,13,16,11,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,3,4,7)(2,5,6,8)(9,11,13,14)(10,15,16,12)(17,19,22,29)(18,23,25,26)(20,27,21,28)(24,31,30,32);(2,6)(5,8)(10,16)(12,15)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(10,16)(12,15)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,32,26,31);(9,12,14,16,13,15,11,10)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310349);