#495302

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	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 592 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,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, 402498);