#7645855

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

6662 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)(6,7)(9,17)(11,13)(15,20)(21,22,26,25)(23,29,31,27)(24,32,33,28)(30,34,36,35);(1,2)(3,4)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35);(5,6,10,7)(8,11,16,13)(9,12,17,14)(15,18,20,19)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36);(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,35,36,34);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=h,e^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, 7553051);