#7718383

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

4652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=i,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, 7625579);