#7655552

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

6859 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)(5,6,11,10)(7,14,9,12)(8,16,18,13)(15,19,17,20);(1,2)(3,4)(21,22)(23,31)(24,28)(25,29)(26,30)(27,33)(32,36)(34,35);(5,7,11,9)(6,12,10,14)(8,15,18,17)(13,19,16,20)(21,23,26,33)(22,27,30,31)(24,25,32,35)(28,34,36,29);(5,8)(6,13)(7,15)(9,17)(10,16)(11,18)(12,19)(14,20)(21,24,26,32)(22,28,30,36)(23,25,33,35)(27,34,31,29);(5,9,11,7)(6,12,10,14)(8,17,18,15)(13,19,16,20)(21,25)(22,29)(23,32)(24,33)(26,35)(27,28)(30,34)(31,36);(5,10,11,6)(7,14,9,12)(8,16,18,13)(15,20,17,19)(21,22)(23,27)(24,28)(25,34)(26,30)(29,35)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,e^2=h,f^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, 7562748);