#7743453

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

6542 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)(5,6,10,11)(7,12,14,9)(8,16,18,13)(15,17,20,19)(21,22,27,26)(23,28,33,31)(24,30,25,29)(32,36,34,35);(1,2,3,4)(21,23)(22,28)(24,34)(25,32)(26,31)(27,33)(29,36)(30,35);(6,11)(7,14)(13,16)(15,20)(21,24,27,25)(22,29,26,30)(23,32,33,34)(28,35,31,36);(5,7)(6,12)(8,15)(9,11)(10,14)(13,19)(16,17)(18,20)(21,25,27,24)(22,30,26,29)(23,32,33,34)(28,35,31,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,23)(22,31)(24,34)(25,32)(26,28)(27,33)(29,35)(30,36);(5,9,10,12)(6,7,11,14)(8,17,18,19)(13,15,16,20)(21,26,27,22)(23,28,33,31)(24,30,25,29)(32,35,34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,e^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=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, 7650649);