#7658202

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

6566 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)(8,12)(9,11)(10,13)(15,20)(16,18)(17,19)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33);(1,2,3,4)(23,30)(26,29)(31,36)(33,35);(5,7,10,9)(6,11,13,14)(8,15,17,16)(12,18,19,20)(21,23,25,30)(22,26,28,29)(24,31,32,36)(27,33,34,35);(5,8,10,17)(6,12,13,19)(7,15,9,16)(11,18,14,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(5,9,10,7)(6,11,13,14)(8,16,17,15)(12,18,19,20)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35);(5,6)(7,11)(8,12)(9,14)(10,13)(15,18)(16,20)(17,19)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565398);