#474924

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382120);