#7726775

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

7336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7633971);