#6405026

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	C₂×#120	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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=efi,c^b=ci,d^b=df,d^c=di

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, 6312222);