#348085

order	512
nilpotency class	3
isoclinism	isoclinic class of #348082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255281);