#537425

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2022 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=fg,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, 444621);