#7634609

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

7002 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)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(2,4)(23,29)(26,33)(30,31)(34,35);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,29)(22,26,28,33)(24,31,32,30)(27,35,36,34);(6,11)(7,14)(12,18)(15,20)(21,24,25,32)(22,27,28,36)(23,30,29,31)(26,34,33,35);(6,11)(10,13)(12,18)(17,19)(22,28)(26,33)(27,36)(34,35);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)

Polycyclic presentation:

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

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, 7541805);