#496703

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 546 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)(13,14)(15,25)(16,24)(17,20)(18,33)(19,22)(21,29)(23,27)(26,40)(28,43)(30,39)(31,35)(32,38)(34,41)(36,44)(37,42);(5,7,8,12)(6,9,10,11)(14,20)(15,26)(22,24)(23,35)(28,30)(29,41)(33,38)(42,44);(7,12)(9,11)(13,15,27,42,19,28,32,41,17,26,31,44,16,30,18,29)(14,21,33,39,24,36,35,40,20,34,38,43,22,37,23,25);(6,10)(9,11)(13,16,17,19)(14,22,20,24)(15,28,26,30)(18,31,32,27)(21,36,34,37)(23,38,35,33)(25,39,40,43)(29,42,41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fh,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^c=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, 403899);