#7972703

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

6669 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)(5,6,11,13)(7,15,16,12)(8,18,17,14)(9,19,20,10)(23,29)(26,33)(30,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,10)(9,14)(11,16)(13,15)(17,19)(18,20)(21,22)(23,26)(24,31)(25,28)(27,32)(29,33)(30,36)(34,35);(6,13)(7,16)(8,17)(9,20)(21,22,25,28)(23,26,29,33)(24,27,32,31)(30,34,35,36);(5,8,11,17)(6,14,13,18)(7,10,16,19)(9,15,20,12)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(5,9,11,20)(6,10,13,19)(7,14,16,18)(8,15,17,12)(21,22)(23,26)(24,31)(25,28)(27,32)(29,33)(30,36)(34,35);(5,10)(6,9)(7,17)(8,16)(11,19)(12,18)(13,20)(14,15)(21,24)(22,27)(23,30)(25,32)(26,34)(28,31)(29,35)(33,36)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,e^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, 7879899);