#7732251

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

6906 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)(5,6,10,11)(7,12,14,9)(8,16,18,13)(15,17,20,19)(21,22,25,27)(23,26,29,34)(24,28,32,35)(30,33,31,36);(1,2,3,4)(21,23)(22,26)(24,31)(25,29)(27,34)(28,36)(30,32)(33,35);(6,11)(7,14)(13,16)(15,20)(22,27)(23,29)(28,35)(30,31);(5,7)(6,12)(8,15)(9,11)(10,14)(13,19)(16,17)(18,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,32)(35,36);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,22)(23,26)(24,28)(25,27)(29,34)(30,33)(31,36)(32,35);(5,9,10,12)(6,7,11,14)(8,17,18,19)(13,15,16,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,35)(29,31)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=hi,f^2=hi,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, 7639447);