#7743460

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

6656 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)(7,12)(8,16)(9,14)(10,11)(13,18)(15,17)(19,20)(22,26)(23,29)(28,31)(30,35);(1,2)(3,4)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(6,11)(7,14)(13,16)(15,20)(21,22,25,26)(23,27,29,33)(24,28,32,31)(30,34,35,36);(5,7,10,14)(6,12,11,9)(8,15,18,20)(13,19,16,17)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,9,10,12)(6,14,11,7)(8,17,18,19)(13,20,16,15)(21,24,25,32)(22,28,26,31)(23,30,29,35)(27,34,33,36)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,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, 7650656);