#7719398

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

5170 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,e^2=hi,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, 7626594);