#7719329

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

4684 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:

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

Polycyclic presentation:

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

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, 7626525);