#7647043

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
prime radical	C₂×#3459	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6764 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)(7,11)(8,16)(9,15)(10,13)(12,18)(14,17)(19,20)(21,22)(23,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(1,2)(3,4)(5,7,10,15)(6,11,13,9)(8,14,18,20)(12,19,16,17)(21,23,27,25)(22,28,24,31)(26,30,34,33)(29,35,32,36);(5,8)(6,12)(7,14)(9,17)(10,18)(11,19)(13,16)(15,20)(21,24,27,22)(23,28,25,31)(26,32,34,29)(30,35,33,36);(5,9)(6,7)(8,17)(10,11)(12,14)(13,15)(16,20)(18,19)(21,25,27,23)(22,28,24,31)(26,33,34,30)(29,35,32,36);(5,6)(7,11)(8,12)(9,15)(10,13)(14,19)(16,18)(17,20)(21,26,27,34)(22,29,24,32)(23,30,25,33)(28,35,31,36);(5,8)(6,12)(7,14)(9,17)(10,18)(11,19)(13,16)(15,20)

Polycyclic presentation:

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

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, 7554239);