#7719980

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

6658 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)(7,12)(8,9)(10,20)(11,14)(13,18)(15,19)(16,17)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7,11,10)(6,12,14,20)(8,17,18,15)(9,16,13,19)(22,26)(23,29)(28,33)(31,35);(5,8,11,18)(6,13,14,9)(7,15,10,17)(12,16,20,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,9,11,13)(6,8,14,18)(7,16,10,19)(12,17,20,15);(5,8)(6,13)(7,15)(9,14)(10,17)(11,18)(12,16)(19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,10,11,7)(6,12,14,20)(8,17,18,15)(9,19,13,16)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627176);