#498056

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

528 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405252);