#580039

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 487235);