#7720259

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

6494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627455);