#508145

order	512
nilpotency class	4
isoclinism	isoclinic class of #508070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415341);