#7720229

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

6527 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)(5,6,11,13)(7,15,9,12)(8,14,18,10)(16,17,19,20)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7,11,9)(6,12,13,15)(8,16,18,19)(10,17,14,20)(22,26)(23,29)(28,33)(31,35);(6,13)(8,18)(12,15)(16,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,8)(6,14)(7,16)(9,19)(10,13)(11,18)(12,20)(15,17);(5,9,11,7)(6,15,13,12)(8,19,18,16)(10,20,14,17)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,10,11,14)(6,8,13,18)(7,17,9,20)(12,16,15,19)(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,e^2=h,f^2=hi,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, 7627425);