#457256

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=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, 364452);