#7824389

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

7196 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)(5,6,10,9)(7,12,14,11)(8,16,18,13)(15,19,17,20)(21,22)(23,27)(24,32)(25,30)(26,28)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,12)(10,14)(13,19)(16,20)(17,18)(21,23,26,25)(22,27,28,30)(24,31,34,33)(29,35,32,36);(6,9)(7,14)(13,16)(15,17)(22,28)(27,30)(29,32)(35,36);(5,7)(6,12)(8,17)(9,11)(10,14)(13,19)(15,18)(16,20)(21,22)(23,27)(24,29)(25,30)(26,28)(31,35)(32,34)(33,36);(5,8)(6,13)(7,15)(9,16)(10,18)(11,19)(12,20)(14,17)(21,24)(22,29)(23,31)(25,33)(26,34)(27,35)(28,32)(30,36);(5,9,10,6)(7,11,14,12)(8,16,18,13)(15,19,17,20)(21,25,26,23)(22,30,28,27)(24,33,34,31)(29,36,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7731585);