#241644

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
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)(27,32)(28,39)(29,44)(30,34)(31,35)(33,42)(36,53)(37,48)(38,49)(40,46)(41,52)(43,51)(45,47)(50,54)(55,56);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,23)(19,21)(20,24)(25,27,31,38)(26,32,35,49)(28,40,41,55)(29,37,30,45)(33,50,51,53)(34,47,44,48)(36,42,54,43)(39,46,52,56);(11,17)(12,21)(14,22)(15,20)(18,19)(23,24)(25,28,31,41)(26,33,35,51)(27,36,38,54)(29,43,30,42)(32,46,49,56)(34,52,44,39)(37,55,45,40)(47,50,48,53);(1,3,4,7)(2,6,5,8)(28,41)(33,51)(36,54)(39,52)(40,55)(42,43)(46,56)(50,53)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,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, 148840);