#7658206

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

6552 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)(7,16)(8,13)(9,12)(10,14)(11,15)(17,18)(19,20)(21,22,26,30)(23,31,32,27)(24,28,25,29)(33,36,34,35);(1,2,3,4)(23,32)(27,31)(33,34)(35,36);(5,7,11,9)(6,12,15,16)(8,17,14,19)(10,18,13,20)(21,23,26,32)(22,27,30,31)(24,33,25,34)(28,35,29,36);(5,8,11,14)(6,13,15,10)(7,17,9,19)(12,20,16,18)(21,22)(23,27)(24,28)(25,29)(26,30)(31,32)(33,35)(34,36);(5,9,11,7)(6,12,15,16)(8,19,14,17)(10,18,13,20)(21,24,26,25)(22,28,30,29)(23,33,32,34)(27,35,31,36);(5,10,11,13)(6,14,15,8)(7,18,9,20)(12,19,16,17)(21,25,26,24)(22,29,30,28)(23,34,32,33)(27,36,31,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7565402);