#348093

order	512
nilpotency class	3
isoclinism	isoclinic class of #348090
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)(20,33)(21,24)(25,36)(27,38)(28,40)(29,44)(30,31)(34,35)(37,42)(39,43)(45,46);(17,18,24,38,22,27,21,23)(19,28,31,36,32,25,30,40)(20,26,33,42,35,41,34,37)(29,43,46,47,45,39,44,48);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,25)(20,29)(21,30)(22,32)(23,36)(24,31)(26,39)(27,28)(33,46)(34,44)(35,45)(37,47)(38,40)(41,43)(42,48);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20)(18,26)(19,29)(21,34)(22,35)(23,37)(24,33)(25,39)(27,41)(28,43)(30,44)(31,46)(32,45)(36,47)(38,42)(40,48)

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,c^b=ch,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, 255289);