#456634

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	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 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)(7,15)(8,20,22,12)(9,24)(10,14)(11,18)(13,25)(16,28)(17,31,33,36)(19,26)(21,29,23,30)(27,32,35,34);(1,2)(3,4)(5,7,9,18,10,19,25,16)(6,11,13,15,14,28,24,26)(8,17,23,34,22,33,21,32)(12,27,30,31,20,35,29,36);(2,4)(7,16)(8,21)(9,25)(11,26)(12,29)(13,24)(15,28)(17,33)(18,19)(20,30)(22,23)(27,35);(5,8)(6,12)(7,17)(9,23)(10,22)(11,27)(13,30)(14,20)(15,31)(16,32)(18,34)(19,33)(21,25)(24,29)(26,36)(28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,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, 363830);