#7642800

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
prime radical	#36717	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7008 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)(5,6,11,9)(7,13,8,12)(10,17,18,14)(15,19,16,20)(24,31)(28,33)(29,35)(34,36);(1,2)(3,4)(5,7,11,8)(6,12,9,13)(10,15,18,16)(14,19,17,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(6,9)(12,13)(14,17)(19,20)(22,26)(27,32)(28,33)(34,36);(5,8,11,7)(6,13,9,12)(10,16,18,15)(14,20,17,19)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9,11,6)(7,13,8,12)(10,17,18,14)(15,20,16,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,10)(6,14)(7,15)(8,16)(9,17)(11,18)(12,19)(13,20)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,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, 7549996);