#254016

order	512
nilpotency class	3
isoclinism	isoclinic class of #253987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161212);