#6402108

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	#2714	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(9,10,13,16)(11,15,12,14)(17,18,24,27,23,29,21,32)(19,30,31,26,28,25,20,22);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,25)(20,21)(22,27)(23,28)(24,31)(26,32)(29,30);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20,23,31)(18,26,29,22)(19,21,28,24)(25,32,30,27);(9,11,13,12)(10,15,16,14)(17,21,23,24)(18,27,29,32)(19,31,28,20)(22,25,26,30);(9,12,13,11)(10,15,16,14)(17,22,24,30,23,26,21,25)(18,28,27,20,29,19,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,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, 6309304);