#475580

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382776);