#7744897

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7652093);