#7903435

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

7210 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)(5,6)(7,13)(8,14)(9,12)(10,11)(15,17)(16,20)(18,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2)(3,4)(5,7)(6,8)(9,15)(10,14)(11,13)(12,16)(17,19)(18,20);(6,11)(7,14)(12,19)(15,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7)(6,8)(9,15)(10,14)(11,13)(12,16)(17,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,7)(9,17)(10,13)(11,14)(12,20)(15,19)(16,18)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7810631);