#7743518

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

6730 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)(5,6,10,12)(7,13,14,8)(9,11,18,20)(15,19,17,16)(21,22)(23,27)(24,31)(25,33)(26,29)(28,34)(30,32)(35,36);(1,2)(3,4)(5,7)(6,8)(9,17)(10,14)(11,19)(12,13)(15,18)(16,20)(21,23,26,31)(22,27,29,24)(25,30,34,36)(28,35,33,32);(7,14)(8,13)(15,17)(16,19)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,36)(34,35);(5,7)(6,8)(9,15)(10,14)(11,16)(12,13)(17,18)(19,20)(21,22)(23,27)(24,31)(25,28)(26,29)(30,35)(32,36)(33,34);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,33)(31,36);(5,9,10,18)(6,11,12,20)(7,15,14,17)(8,16,13,19)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650714);