#7651832

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

6812 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)(5,6,10,9)(7,13,15,11)(8,12,18,17)(14,20,16,19);(1,2)(3,4)(21,22,26,27)(23,28,32,35)(24,25,29,34)(30,36,33,31);(6,9)(11,13)(12,17)(19,20)(22,27)(25,34)(28,35)(31,36);(5,7)(6,11)(8,16)(9,13)(10,15)(12,20)(14,18)(17,19)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,30)(34,36);(5,8,10,18)(6,12,9,17)(7,14,15,16)(11,19,13,20)(21,24,26,29)(22,25,27,34)(23,30,32,33)(28,36,35,31);(5,9,10,6)(7,13,15,11)(8,17,18,12)(14,20,16,19)(21,25)(22,29)(23,31)(24,27)(26,34)(28,30)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559028);