#251958

order	512
nilpotency class	3
isoclinism	isoclinic class of #251935
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,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, 159154);