#7709274

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
prime radical	C₂×#3542	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7086 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)(7,14)(8,9)(10,12)(11,15)(13,18)(16,20)(17,19)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,14)(13,19)(18,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(6,12)(7,15)(8,13)(17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,8,10,13)(6,9,12,18)(7,16,15,19)(11,17,14,20);(5,7)(6,11)(8,16)(9,17)(10,15)(12,14)(13,19)(18,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9)(6,13)(7,17)(8,12)(10,18)(11,19)(14,16)(15,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)

Polycyclic presentation:

2^9|b^2=hi,d^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, 7616470);