#499675

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,28,36)(20,24,33,41)(21,25,32,40)(27,35,44,47)(29,37,34,42)(30,38,43,46)(31,39,45,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(19,27)(20,30)(21,32)(23,35)(24,38)(25,40)(28,44)(29,45)(31,34)(33,43)(36,47)(37,48)(39,42)(41,46);(10,14)(13,15)(17,19,22,28)(18,23,26,36)(20,31,33,45)(21,34,32,29)(24,39,41,48)(25,42,40,37)(27,43,44,30)(35,46,47,38);(1,4,5,3)(2,6,7,8)(19,28)(23,36)(27,44)(29,34)(31,45)(35,47)(37,42)(39,48)

Polycyclic presentation:

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

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, 406871);