#251968

order	512
nilpotency class	3
isoclinism	isoclinic class of #251967
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	#9980	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

889 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 126 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)(19,27)(20,24)(21,34)(22,26)(23,31)(25,37)(28,35)(29,45)(30,41)(32,42)(33,40)(36,39)(38,47)(43,46)(44,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,18)(19,28,30,23)(20,24)(21,25)(22,26)(27,31,41,35)(29,44,46,38)(32,39)(33,40)(34,37)(36,42)(43,48,45,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,23)(20,29)(21,35)(22,30)(24,38)(25,41)(26,28)(27,34)(31,37)(32,48)(33,46)(36,47)(39,43)(40,44)(42,45);(4,11)(7,15)(9,16)(12,14)(17,20,22,33)(18,24,26,40)(19,29,30,46)(21,36,37,32)(23,38,28,44)(25,42,34,39)(27,43,41,45)(31,48,35,47)

Polycyclic presentation:

2^9|b^2=e,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,d^c=di,e^c=eh,f^c=fh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159164);