#553968

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

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, 461164);