#463633

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

978 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)(6,9)(7,11)(13,14,17,19)(15,20,16,18)(21,22,26,29)(23,32,24,35)(25,31,27,30)(28,34,36,33);(5,6,8,9)(7,10,11,12)(21,23,25,33,26,24,27,34)(22,28,30,32,29,36,31,35);(5,7,8,11)(6,10,9,12)(13,15,17,16)(14,18,19,20)(22,29)(23,33)(24,34)(25,27)(28,35)(32,36);(2,4)(7,11)(10,12)(13,16,17,15)(14,18,19,20)(21,24,25,34,26,23,27,33)(22,28,30,32,29,36,31,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=fi,d^2=efi,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, 370829);