#454682

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1829 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 361878);