#458067

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365263);