#538886

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2140 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)(5,6,10,9)(7,12,8,11)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(13,14,17,16)(15,19,20,18)(22,26)(27,29)(28,32)(35,36);(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,18,16,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(1,3,4,2)(5,8,10,7)(6,12,9,11)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33);(5,9,10,6)(7,12,8,11)(13,16,17,14)(15,19,20,18)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 446082);