#475653

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

674 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,21,25)(19,28,31,23)(20,33,35,43)(22,27,29,26)(24,37,42,36)(30,41,46,48)(32,39,45,40)(34,44,47,38);(10,13)(14,15)(17,19,29,45,21,31,22,32)(18,23,26,39,25,28,27,40)(20,34,37,30,35,47,36,46)(24,41,33,38,42,48,43,44);(1,3,4,7)(2,5,6,8)(11,16)(14,15)(17,20,21,35)(18,24,25,42)(19,30,31,46)(22,37,29,36)(23,38,28,44)(26,43,27,33)(32,47,45,34)(39,41,40,48);(2,6)(5,8)(9,11,12,16)(10,14,13,15)(20,35)(24,42)(30,46)(33,43)(34,47)(36,37)(38,44)(41,48)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=eh,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, 382849);