#7652162

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6310 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)(21,22)(23,28)(24,29)(25,30)(26,31)(27,32)(33,35)(34,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,24)(22,28,32,29)(25,31,35,34)(26,33,36,30);(6,10)(11,17)(12,18)(19,20)(21,23,27,24)(22,29,32,28)(25,31,35,34)(26,30,36,33);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,27,23)(22,28,32,29)(25,31,35,34)(26,30,36,33);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,35)(22,30,32,33)(23,31,24,34)(26,29,36,28);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,26)(22,31)(23,33)(24,30)(25,29)(27,36)(28,35)(32,34)

Polycyclic presentation:

2^9|a^2=g,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, 7559358);