#460095

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1182 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)(7,17,20,12)(8,22,24,30)(9,16)(10,14)(11,15)(13,25,23,26)(18,27,29,21)(19,32)(28,35)(31,34)(33,36);(1,2)(3,4)(5,7)(6,12,16,17)(8,19)(9,20)(10,21)(11,27)(13,28,23,31)(14,29,15,18)(22,32,30,36)(24,33)(25,34)(26,35);(2,4)(5,6)(7,18,20,29)(8,23,24,13)(9,16)(10,14)(11,15)(12,27,17,21)(19,32)(22,25,30,26)(28,35)(31,34)(33,36);(5,8,9,24)(6,13)(7,19,20,33)(10,26,11,25)(12,28)(14,30)(15,22)(16,23)(17,31)(18,32)(21,35,27,34)(29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,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, 367291);