#7709649

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

6906 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)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(6,11)(7,12)(13,18)(14,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616845);