#496706

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fhi,d^2=i,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, 403902);