#7736703

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

6865 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,27,31)(23,32,24,28)(25,29,35,34)(26,30,36,33);(1,2,3,4)(21,23,27,24)(22,28,31,32)(25,30,35,33)(26,34,36,29);(6,10)(11,13)(12,17)(18,19)(23,24)(25,35)(28,32)(29,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,27,23)(22,28,31,32)(25,33,35,30)(26,34,36,29);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,25)(22,29)(23,30)(24,33)(26,28)(27,35)(31,34)(32,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26)(22,30)(23,29)(24,34)(25,28)(27,36)(31,33)(32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643899);