#7763443

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6750 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)(7,13)(8,16)(9,11)(10,14)(12,18)(15,19)(17,20)(21,22)(23,28)(24,25)(26,31)(27,29)(30,36)(32,33)(34,35);(1,2)(3,4)(5,7)(6,10)(8,17)(9,14)(11,13)(12,20)(15,18)(16,19)(21,23,27,26)(22,28,29,31)(24,32,30,34)(25,33,36,35);(6,11)(7,14)(12,16)(15,17)(22,29)(24,30)(28,31)(32,34);(5,6)(7,13)(8,12)(9,11)(10,14)(15,20)(16,18)(17,19)(21,24,27,30)(22,25,29,36)(23,32,26,34)(28,33,31,35);(5,8,9,18)(6,12,11,16)(7,15,14,17)(10,19,13,20)(21,25)(22,30)(23,33)(24,29)(26,35)(27,36)(28,34)(31,32);(5,7)(6,13)(8,15)(9,14)(10,11)(12,20)(16,19)(17,18)(21,26,27,23)(22,31,29,28)(24,34,30,32)(25,35,36,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7670639);