#7739314

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

6686 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)(21,22)(23,30)(24,28)(25,27)(26,29)(31,36)(32,35)(33,34);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(6,10)(7,13)(12,15)(14,19)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,26,27,34)(22,29,25,33)(23,31,24,32)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7646510);