#251959

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 822 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) = 128

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=h,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, 159155);