#521708

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2516 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=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, 428904);