#7714955

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

6423 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=h,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, 7622151);