#503512

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

836 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410708);