#7876766

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

6834 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7783962);