#495306

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fhi,d^2=ei,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, 402502);