#7657300

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

6654 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564496);