#456645

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 363841);