#508044

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1198 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 166 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)(13,14,18,22)(15,23,26,42)(16,27,31,41)(17,32,30,21)(19,34,36,24)(20,37,39,28)(25,43,33,38)(29,40,44,35);(6,9)(10,12)(15,24)(16,28)(17,30)(19,23)(20,27)(21,32)(25,44)(26,34)(29,33)(31,37)(35,43)(36,42)(38,40)(39,41);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,19,22,36)(16,29,31,44)(17,33,30,25)(20,38,39,43)(21,40,32,35)(23,41,42,27)(24,37,34,28);(7,11)(10,12)

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=de,f^c=fh,g^c=gh,h^c=hi

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, 415240);