#477163

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1105 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6,9,8)(7,10,11,12)(13,14)(15,25)(16,21)(17,23)(18,24)(19,35)(20,28)(22,32)(26,37)(27,40)(29,39)(30,44)(31,41)(33,38)(34,42)(36,43);(1,3,4,2)(6,8)(10,12)(13,15)(14,20)(16,30)(17,28)(18,33)(19,29)(21,40)(22,36)(23,25)(24,39)(26,32)(27,31)(34,43)(35,38)(37,42)(41,44);(2,3)(5,7,9,11)(6,10,8,12)(13,16,17,31)(14,21,23,41)(15,26,28,43)(18,34,19,32)(20,36,25,37)(22,24,42,35)(27,33,30,29)(38,44,39,40);(2,3)(5,8,9,6)(7,10,11,12)(13,16)(14,22)(15,27)(17,31)(18,32)(19,34)(20,37)(21,35)(23,42)(24,41)(25,36)(26,29)(28,30)(33,43)(38,44)(39,40)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^d=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, 384359);