#7657700

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

6729 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)(5,6,12,11)(7,16,14,9)(8,13,20,19)(10,15,18,17)(21,22,26,29)(23,30,33,27)(24,25,28,34)(31,36,35,32);(1,2)(3,4)(5,7)(6,9)(8,15)(10,13)(11,16)(12,14)(17,20)(18,19);(5,8)(6,13)(7,17)(9,18)(10,16)(11,19)(12,20)(14,15)(21,23,26,33)(22,27,29,30)(24,31,28,35)(25,32,34,36);(5,9,12,16)(6,14,11,7)(8,18,20,10)(13,15,19,17);(5,10,12,18)(6,15,11,17)(7,13,14,19)(8,16,20,9)(21,24)(22,25)(23,31)(26,28)(27,32)(29,34)(30,36)(33,35);(5,11,12,6)(7,16,14,9)(8,19,20,13)(10,15,18,17)(21,25)(22,28)(23,32)(24,29)(26,34)(27,35)(30,31)(33,36)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,e^2=i,f^2=i,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, 7564896);