#7642570

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

7246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7549766);