#7726829

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

6816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,11,12)(7,14,8,10)(9,17,18,13)(15,19,16,20)(21,22,26,24)(23,27,29,31)(25,28,33,32)(30,34,35,36);(1,2,3,4)(5,7,11,8)(6,10,12,14)(9,15,18,16)(13,19,17,20)(23,29)(25,33)(27,31)(28,32);(6,12)(7,8)(9,18)(19,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(5,8,11,7)(6,10,12,14)(9,16,18,15)(13,19,17,20);(5,9)(6,13)(7,15)(8,16)(10,19)(11,18)(12,17)(14,20)(21,24,26,22)(23,27,29,31)(25,32,33,28)(30,34,35,36);(5,10)(6,7)(8,12)(9,19)(11,14)(13,15)(16,17)(18,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7634025);