#7649161

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

5626 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7556357);