#7644314

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	C₂×#3493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6550 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)(7,8)(10,18)(12,15)(13,17)(23,29)(24,32)(26,30)(27,34);(1,2)(3,4)(5,6,11,9)(7,12,8,15)(10,13,18,17)(14,19,16,20)(21,22,25,28)(23,30,29,26)(24,27,32,34)(31,36,35,33);(5,7,11,8)(6,12,9,15)(10,14,18,16)(13,19,17,20)(21,23,25,29)(22,26,28,30)(24,31,32,35)(27,33,34,36);(5,8,11,7)(6,12,9,15)(10,16,18,14)(13,19,17,20)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36);(5,9,11,6)(7,12,8,15)(10,17,18,13)(14,19,16,20);(5,10)(6,13)(7,14)(8,16)(9,17)(11,18)(12,19)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551510);