#241666

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,23)(12,20,21,15)(17,24,19,22)(25,26)(27,36)(28,40)(29,34)(30,45)(31,35)(32,38)(33,43)(37,50)(39,48)(41,53)(42,52)(44,51)(46,49)(47,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,23)(19,21)(20,24)(25,27,29,38)(26,32,34,36)(28,41,42,55)(30,39,31,46)(33,50,51,56)(35,48,45,49)(37,43,54,44)(40,47,52,53);(11,17)(12,21)(14,22)(15,20)(18,19)(23,24)(25,28,29,42)(26,33,34,51)(27,37,38,54)(30,44,31,43)(32,47,36,53)(35,52,45,40)(39,55,46,41)(48,50,49,56);(1,3,4,7)(2,6,5,8)(27,38)(32,36)(37,54)(39,46)(41,55)(47,53)(48,49)(50,56)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148862);