#7709944

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

7004 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)(8,10)(9,17)(11,13)(14,18)(15,19)(16,20)(24,31)(28,33)(29,35)(34,36);(1,2)(3,4)(5,7,11,8)(6,12,13,10)(9,15,18,16)(14,20,17,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(6,13)(7,8)(9,18)(19,20)(22,26)(27,32)(28,33)(34,36);(5,8,11,7)(6,10,13,12)(9,16,18,15)(14,19,17,20)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9)(6,14)(7,15)(8,16)(10,19)(11,18)(12,20)(13,17)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,10)(6,7)(8,13)(9,19)(11,12)(14,15)(16,17)(18,20)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|b^2=h,d^2=i,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, 7617140);