#7711819

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

6786 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)(9,17)(12,16)(14,20)(18,19)(21,22)(23,28)(24,27)(25,33)(26,30)(29,34)(31,35)(32,36);(1,2)(3,4)(5,6,10,8)(7,11,15,13)(9,12,17,16)(14,18,20,19)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(6,8)(11,13)(12,16)(18,19)(21,22,26,30)(23,28,24,27)(25,29,34,33)(31,36,32,35);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,36)(33,34);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)

Polycyclic presentation:

2^9|b^2=i,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^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, 7619015);