#7720540

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

6980 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,26,27)(23,25,29,35)(24,31,33,28)(30,36,32,34);(1,2,3,4)(21,23)(22,25)(24,32)(26,29)(27,35)(28,36)(30,33)(31,34);(6,10)(11,13)(12,17)(18,19)(22,27)(23,29)(24,33)(34,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,25)(24,30)(26,29)(27,35)(28,34)(31,36)(32,33);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24)(22,28)(23,30)(25,34)(26,33)(27,31)(29,32)(35,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25)(22,23)(24,34)(26,35)(27,29)(28,30)(31,32)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7627736);