#348097

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
prime radical	#3531	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 729 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 255293);