#382509

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2368 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 2368 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 128

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

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

Polycyclic presentation:

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

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, 289705);