#7651038

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

6732 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)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22,27,31)(23,32,35,28)(24,34,33,29)(25,30,36,26);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20);(7,13)(11,15)(14,19)(18,20)(21,23,27,35)(22,28,31,32)(24,25,33,36)(26,29,30,34);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,24)(22,29)(23,25)(26,32)(27,33)(28,30)(31,34)(35,36);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(21,25)(22,30)(23,33)(24,35)(26,31)(27,36)(28,34)(29,32);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,26,27,30)(22,25,31,36)(23,34,35,29)(24,32,33,28)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,e^2=i,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, 7558234);