#497135

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 710 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)(21,27)(22,32)(23,35)(24,39)(28,37)(30,43)(31,36)(34,44)(38,41)(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)(26,35,37,29)(27,36,39,33)(30,34,42,38)(40,44,43,41);(5,8,10,9)(6,11,7,12)(13,16,27,41,19,30,31,35,17,28,39,44,32,42,18,29)(14,20,33,40,22,34,24,37,15,23,36,43,25,38,21,26);(5,9,10,8)(6,11,7,12)(14,15)(20,23)(21,24)(22,25)(26,37)(33,36)(34,38)(40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efhi,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, 404331);