#7711389

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

7012 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)(5,6)(7,12)(8,11)(9,17)(10,14)(13,18)(15,19)(16,20)(24,32)(28,33)(30,36)(34,35);(1,2)(3,4)(5,7,10,8)(6,11,14,12)(9,15,18,16)(13,19,17,20)(22,26)(27,29)(28,33)(34,35);(5,6,10,14)(7,12,8,11)(9,13,18,17)(15,20,16,19)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,6)(7,11)(8,12)(9,13)(10,14)(15,19)(16,20)(17,18);(5,8,10,7)(6,12,14,11)(9,16,18,15)(13,20,17,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,9)(6,13)(7,15)(8,16)(10,18)(11,19)(12,20)(14,17)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618585);