#7642799

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

7000 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)(7,14)(8,12)(9,10)(11,13)(15,20)(16,19)(17,18)(24,31)(28,33)(29,35)(34,36);(1,2)(3,4)(5,7,11,8)(6,12,13,14)(9,16,15,18)(10,17,20,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(6,13)(9,15)(12,14)(16,18)(22,26)(27,32)(28,33)(34,36);(5,8,11,7)(6,14,13,12)(9,18,15,16)(10,19,20,17)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9,11,15)(6,10,13,20)(7,16,8,18)(12,17,14,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,10)(6,15)(7,17)(8,19)(9,13)(11,20)(12,18)(14,16)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549995);