#7714643

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

6634 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)(7,8,13,16)(9,12,17,18)(14,15,19,20)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(1,2,3,4)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(6,11)(7,13)(12,18)(14,19)(23,24)(27,30)(31,32)(35,36);(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,19)(18,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,9,10,17)(6,12,11,18)(7,14,13,19)(8,15,16,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,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, 7621839);