#6385603

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#2714	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=df,d^c=di,e^c=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, 6292799);