#333195

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256

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

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

Polycyclic presentation:

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

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, 240391);