#7652438

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6395 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)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22)(23,28)(24,29)(25,30)(26,31)(27,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,27,24)(22,28,32,29)(25,31,35,34)(26,33,36,30);(7,13)(11,14)(15,19)(18,20)(21,23,27,24)(22,29,32,28)(25,31,35,34)(26,30,36,33);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,24,27,23)(22,28,32,29)(25,31,35,34)(26,30,36,33);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,27,35)(22,30,32,33)(23,31,24,34)(26,29,36,28);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,26)(22,31)(23,33)(24,30)(25,29)(27,36)(28,35)(32,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559634);