#7715653

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

6882 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)(7,11)(8,10)(9,16)(12,17)(13,14)(15,19)(18,20)(25,33)(29,34)(30,36)(32,35);(1,2)(3,4)(5,6,10,8)(7,13,14,11)(9,12,17,16)(15,19,20,18)(21,22)(23,24)(25,29)(26,27)(28,31)(30,32)(33,34)(35,36);(6,8)(7,14)(12,16)(15,20)(22,27)(24,28)(29,34)(32,35);(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,19,20,18)(21,24,26,28)(22,23,27,31)(25,32,33,35)(29,30,34,36);(5,9,10,17)(6,12,8,16)(7,15,14,20)(11,18,13,19)(21,25,26,33)(22,29,27,34)(23,30,31,36)(24,32,28,35)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=i,f^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, 7622849);