#461922

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(21,22,26,29)(23,33,35,30)(24,36,34,28)(25,32,27,31);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,27,24,26,35,25,34)(22,28,32,30,29,36,31,33);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(22,29)(23,24)(25,27)(28,33)(30,36)(34,35);(2,4)(21,24,25,23,26,34,27,35)(22,30,31,28,29,33,32,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=fi,d^2=e,e^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,d^c=de,e^c=ei,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, 369118);