#496409

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

866 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,28,23,29,20,26)(19,24,32,40,33,41,30,38)(21,25,34,42,35,27,36,43)(31,37,44,47,45,39,46,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,33)(18,24,29,41)(20,30,22,32)(21,31,35,45)(25,37,27,39)(26,38,28,40)(34,44,36,46)(42,47,43,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(20,22)(21,35)(24,37)(26,43)(27,29)(28,42)(30,32)(31,45)(38,48)(39,41)(40,47);(2,8)(6,9)(7,16)(14,15)(17,20,23,22)(18,26,29,28)(19,30,33,32)(21,34,35,36)(24,38,41,40)(25,42,27,43)(31,44,45,46)(37,47,39,48)

Polycyclic presentation:

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

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, 403605);