#251388

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

834 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=hi,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158584);