#493901

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401097);