#6378451

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

1424 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)(17,18,23,27,22,26,19,24)(20,29,31,32,30,25,28,21);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,26,27)(20,28,30,31)(21,25,32,29);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,31)(24,32)(26,29);(2,5)(6,8)(20,30)(21,32)(25,29)(28,31);(17,21,23,29,22,32,19,25)(18,20,27,31,26,30,24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^c=di,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, 6285647);