#456637

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
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	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1423 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)(7,10)(8,13)(9,14,25,31)(11,15)(12,16)(17,21)(18,24,32,26)(19,27,34,36)(20,28)(22,29)(23,33,35,30);(1,2)(3,4)(5,7,11,16,12,15,10,6)(8,13,21,28,22,29,20,17)(9,18,24,31,25,32,26,14)(19,27,23,30,34,36,35,33);(2,4)(5,8,12,22)(6,13,16,29)(7,17,15,28)(9,23,25,35)(10,21,11,20)(14,30,31,33)(18,27,32,36)(19,26,34,24);(5,9)(6,14)(7,18)(8,19)(10,26)(11,24)(12,25)(13,27)(15,32)(16,31)(17,33)(20,35)(21,23)(22,34)(28,30)(29,36)

Polycyclic presentation:

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

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, 363833);