#7657004

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

6700 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)(5,6,10,14)(7,12,17,11)(8,9,13,19)(15,18,20,16)(21,22)(23,32)(24,30)(25,26)(27,29)(28,31)(33,36)(34,35);(1,2,3,4)(21,23,28,27)(22,29,31,32)(24,35,26,33)(25,34,30,36);(5,7)(6,11)(8,15)(9,16)(10,17)(12,14)(13,20)(18,19)(21,24,28,26)(22,30,31,25)(23,33,27,35)(29,34,32,36);(5,8,10,13)(6,9,14,19)(7,15,17,20)(11,16,12,18)(21,25)(22,24)(23,34)(26,31)(27,36)(28,30)(29,35)(32,33);(5,7)(6,12)(8,15)(9,18)(10,17)(11,14)(13,20)(16,19)(21,26,28,24)(22,25,31,30)(23,33,27,35)(29,34,32,36);(5,9)(6,13)(7,16)(8,14)(10,19)(11,20)(12,15)(17,18)(21,27,28,23)(22,29,31,32)(24,35,26,33)(25,36,30,34)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=i,f^2=i,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, 7564200);