#7647211

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=i,e^2=i,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, 7554407);