#7655382

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

6962 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)(6,11)(7,13)(14,19)(15,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,33)(34,36);(1,2)(3,4)(21,23,26,28)(22,25,29,30)(24,32,34,31)(27,35,36,33);(5,6)(7,12)(8,13)(9,14)(10,11)(15,17)(16,20)(18,19)(21,24,26,34)(22,27,29,36)(23,31,28,32)(25,33,30,35);(5,7,10,13)(6,12,11,8)(9,15,18,20)(14,17,19,16);(5,8,10,12)(6,13,11,7)(9,17,18,16)(14,15,19,20)(21,22)(23,25)(24,27)(26,29)(28,30)(31,33)(32,35)(34,36);(5,9)(6,14)(7,15)(8,16)(10,18)(11,19)(12,17)(13,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,32)(31,36)(34,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562578);