#7652323

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

6458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559519);