#7972979

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

6547 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)(7,14)(8,9)(10,12)(11,15)(13,18)(16,17)(19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,19)(10,15)(12,14)(13,17)(18,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(6,12)(7,15)(8,18)(17,19)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,8)(6,13)(7,16)(9,12)(10,18)(11,17)(14,19)(15,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,7)(6,11)(8,16)(9,19)(10,15)(12,14)(13,17)(18,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9,10,13)(6,8,12,18)(7,17,15,19)(11,20,14,16)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7880175);