#7720313

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

6757 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)(5,6,9,10)(7,11,13,17)(8,15,16,12)(14,20,19,18)(21,22,26,29)(23,24,27,31)(25,33,34,28)(30,36,35,32);(1,2)(3,4)(5,6,9,10)(7,11,13,17)(8,12,16,15)(14,18,19,20)(23,27)(24,31)(30,35)(32,36);(6,10)(7,13)(12,15)(14,19)(21,23)(22,24)(25,30)(26,27)(28,32)(29,31)(33,36)(34,35);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,35)(30,33)(34,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,25,26,34)(22,28,29,33)(23,30,27,35)(24,32,31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=i,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, 7627509);