#7714907

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

6494 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)(8,16)(9,10)(11,15)(12,17)(14,18)(19,20)(22,26)(24,32)(27,29)(30,36);(1,2)(3,4)(6,10)(11,13)(12,16)(18,19);(5,6,9,10)(7,13,15,11)(8,12,17,16)(14,19,20,18)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622103);