#508308

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415504);