#582134

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

2654 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,e^2=gh,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, 489330);