#7709387

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

6948 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)(5,6,11,13)(7,14,9,12)(8,10,17,20)(15,19,18,16)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(5,7,11,9)(6,12,13,14)(8,15,17,18)(10,16,20,19)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(6,13)(7,9)(8,17)(16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,8)(6,10)(7,15)(9,18)(11,17)(12,16)(13,20)(14,19);(5,9,11,7)(6,14,13,12)(8,18,17,15)(10,19,20,16)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,10)(6,8)(7,16)(9,19)(11,20)(12,15)(13,17)(14,18)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,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, 7616583);