#495305

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402501);