#7646306

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

7140 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)(7,14)(8,11)(9,17)(10,13)(12,18)(15,19)(16,20)(21,22,26,30)(23,29,25,27)(24,28,34,32)(31,36,33,35);(1,2,3,4)(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,34)(33,36);(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,19)(17,18)(21,23,26,25)(22,27,30,29)(24,31,34,33)(28,35,32,36);(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,22)(23,27)(24,32)(25,29)(26,30)(28,34)(31,36)(33,35);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,36)(30,32);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,25,26,23)(22,29,30,27)(24,33,34,31)(28,36,32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553502);