#511631

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,29)(28,38)(30,44)(31,36)(32,37)(33,43)(34,48)(35,40)(39,45)(41,47)(42,46);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(21,33)(22,32)(24,38)(25,41)(27,39)(29,45)(30,46)(31,47)(34,35)(36,43)(40,42)(44,48);(3,7)(6,8)(10,14)(13,16)(17,19)(18,24)(20,29)(21,34)(22,35)(23,27)(25,42)(26,44)(28,33)(30,32)(31,46)(36,48)(37,40)(38,41)(39,43)(45,47);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(19,29)(24,27)(28,45)(30,35)(34,46)(38,39)(40,44)(42,48)

Polycyclic presentation:

2^9|a^2=ei,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,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, 418827);