#497133

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#2246	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 766 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404329);