#7712489

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

6520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619685);