#7732849

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

7076 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)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18);(1,2)(3,4)(5,7,11,8)(6,12,9,13)(10,16,18,15)(14,20,17,19)(21,22)(23,28)(24,29)(25,33)(26,27)(30,34)(31,32)(35,36);(6,9)(7,8)(14,17)(15,16)(22,27)(24,28)(30,33)(32,35);(5,8,11,7)(6,13,9,12)(10,16,18,15)(14,20,17,19)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,35,33,32);(5,9,11,6)(7,13,8,12)(10,17,18,14)(15,20,16,19)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,10,11,18)(6,14,9,17)(7,15,8,16)(12,19,13,20)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

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

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, 7640045);