#244089

order	512
nilpotency class	3
isoclinism	isoclinic class of #238015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

480 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 164 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,6)(5,8)(7,11)(10,15)(12,14)(13,16)(17,18,21,25,22,26,28,37)(19,27,30,39,31,42,36,23)(20,32,35,46,34,24,40,41)(29,43,44,45,33,47,48,38);(2,6)(3,10)(5,12)(7,15)(9,16)(11,13)(17,19,28,30,22,31,21,36)(18,23,37,27,26,39,25,42)(20,33,40,48,34,29,35,44)(24,38,46,43,32,45,41,47);(1,3,4,11)(2,7,6,9)(5,13,12,10)(8,15,14,16)(17,20,21,35,22,34,28,40)(18,24,25,41,26,32,37,46)(19,29,30,44,31,33,36,48)(23,38,27,43,39,45,42,47);(3,11)(7,9)(10,13)(15,16)(20,34)(24,32)(29,33)(35,40)(38,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151285);