#511545

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418741);