#508128

order	512
nilpotency class	4
isoclinism	isoclinic class of #508024
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

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, 415324);