#6393331

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1631 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,19,27)(20,25,30,29,31,21,26,32);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,28,27)(20,30,31,26)(21,29,25,32);(10,14)(13,16)(17,20,23,30,22,31,19,26)(18,25,24,29,28,21,27,32);(1,4,6,5)(2,7,3,8)(17,21,22,25)(18,26,28,30)(19,29,23,32)(20,27,31,24);(1,5,6,4)(2,7,3,8)(17,19,22,23)(18,27,28,24)(20,30,31,26)(21,32,25,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,e^c=ei,e^d=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, 6300527);