#508149

order	512
nilpotency class	4
isoclinism	isoclinic class of #508078
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×Q₈	C₂⁴
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 815 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)(19,27)(20,31)(21,35)(22,24)(23,29)(25,34)(26,36)(28,45)(30,41)(32,40)(33,42)(37,48)(38,44)(39,47)(43,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(19,28)(21,36)(23,39)(26,35)(27,45)(29,47)(30,37)(32,46)(33,38)(40,43)(41,48)(42,44);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19)(18,23)(20,32)(21,37)(22,30)(24,40)(25,41)(26,43)(27,31)(28,36)(29,34)(33,47)(35,39)(38,46)(42,45)(44,48);(4,12)(7,14)(10,16)(13,15)(18,24)(23,40)(25,31)(26,35)(27,41)(39,43)(42,44)(45,48)

Polycyclic presentation:

2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 415345);