#6405053

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,27,22,21,26,32)(19,24,30,25,28,29,20,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,22,28)(18,24,21,29)(20,23,30,26)(25,32,31,27);(2,6)(5,7)(9,12,13,11)(10,14,15,16)(17,20,22,30)(18,25,21,31)(19,26,28,23)(24,27,29,32);(3,8)(5,7)(18,21)(19,28)(20,30)(27,32);(2,6)(5,7)(17,21,23,32,22,18,26,27)(19,29,30,31,28,24,20,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=dfi,d^c=di,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, 6312249);