#7662428

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

6817 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)(5,6,10,13)(7,11,14,18)(8,9,12,17)(15,16,19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,34)(33,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20);(7,14)(11,18)(15,19)(16,20)(21,23,27,25)(22,28,24,30)(26,31,34,33)(29,35,32,36);(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,24,27,22)(23,28,25,30)(26,32,34,29)(31,35,33,36);(5,8)(6,9)(7,15)(10,12)(11,16)(13,17)(14,19)(18,20)(21,25,27,23)(22,28,24,30)(26,31,34,33)(29,36,32,35);(5,9)(6,12)(7,16)(8,13)(10,17)(11,19)(14,20)(15,18)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7569624);