#7641678

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6956 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)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,29)(25,30)(26,31)(27,33)(32,36)(34,35);(2,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,33)(22,27,31,28)(24,34,25,32)(29,36,30,35);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,28)(24,32)(25,34)(26,33)(27,31)(29,36)(30,35);(6,11)(10,13)(12,18)(17,19)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,28,36);(7,14)(10,13)(15,20)(17,19)(21,25,26,24)(22,30,31,29)(23,32,33,34)(27,35,28,36);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548874);