#7647149

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=hi,e^2=hi,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, 7554345);