#458079

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 365275);