#7649900

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

7013 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)(5,6,10,13)(7,11,14,18)(8,9,12,17)(15,16,19,20)(22,27)(23,28)(29,33)(30,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,22,26,27)(23,24,28,32)(25,29,34,33)(30,31,35,36);(7,14)(11,18)(15,19)(16,20)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(5,8)(6,9)(7,15)(10,12)(11,16)(13,17)(14,19)(18,20)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(5,9)(6,12)(7,16)(8,13)(10,17)(11,19)(14,20)(15,18)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)

Polycyclic presentation:

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

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, 7557096);