#7646752

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

6756 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)(6,11)(7,14)(8,13)(16,20)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(5,6,10,11)(7,12,14,18)(8,9,13,17)(15,16,19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,7)(6,12)(8,15)(9,16)(10,14)(11,18)(13,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,8)(6,9)(7,15)(10,13)(11,17)(12,16)(14,19)(18,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,9)(6,13)(7,16)(8,11)(10,17)(12,19)(14,20)(15,18)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,7)(6,12)(8,15)(9,16)(10,14)(11,18)(13,19)(17,20)

Polycyclic presentation:

2^9|b^2=h,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, 7553948);