#6402528

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1415 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)(10,14)(11,12)(17,18,23,27,22,25,28,31)(19,21,20,30,29,32,26,24);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,12,16)(17,19,22,29)(18,21,25,32)(20,28,26,23)(24,27,30,31);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,28)(21,27)(22,26)(23,29)(25,30)(31,32);(10,14)(15,16)(18,25)(19,29)(20,26)(27,31);(9,12,13,11)(10,16,14,15)(17,21,23,30,22,32,28,24)(18,26,27,19,25,20,31,29)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6309724);