#544725

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1998 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,15)(16,19,20,18)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(6,9)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(6,9)(10,11)(14,15)(18,19)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,25,27,22)(23,30,24,28)(26,33,34,29)(31,36,32,35);(5,7,8,12)(6,10,9,11)(13,16,17,20)(14,18,15,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

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

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, 451921);