#234446

order	512
nilpotency class	3
isoclinism	isoclinic class of #234438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1299 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 190 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)(9,13)(10,12)(11,14)(15,16)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,44)(32,43)(33,41)(34,40)(35,42)(36,38)(37,47)(39,45)(46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,39)(26,29)(27,43)(28,38)(33,45)(34,47)(35,46)(37,42)(40,44)(41,48);(4,11)(6,12)(7,14)(9,15)(17,20,23,35)(18,25,28,42)(19,30,32,46)(21,34,22,33)(24,37,38,39)(26,41,27,40)(29,44,43,48)(31,47,36,45);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,35,34)(24,29,38,43)(25,40,42,41)(30,45,46,47)(37,48,39,44)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi,e^c=eh

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, 141642);