#6403070

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	#2733	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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)(11,14)(12,16)(13,15)(17,18,24,25,23,27,21,30)(19,28,29,31,20,22,26,32);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,16,13,15)(17,19)(18,22)(20,23)(21,26)(24,29)(25,32)(27,28)(30,31);(2,6)(5,8)(9,12,14,13)(10,15,11,16)(17,20,23,19)(18,22,27,28)(21,29,24,26)(25,32,30,31);(9,12,14,13)(10,16,11,15)(17,21,23,24)(18,25,27,30)(19,29,20,26)(22,31,28,32);(9,13,14,12)(10,16,11,15)(17,22,21,31,23,28,24,32)(18,26,30,20,27,29,25,19)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=fh,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, 6310266);