#458068

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

Polycyclic presentation:

2^9|b^2=ef,c^2=efi,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, 365264);