#190121

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

964 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,36,44,41)(29,42,46,37)(32,39)(34,38)(35,40)(43,47,45,48);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,37)(26,27)(31,41)(32,43)(33,44)(34,45)(35,46)(38,47)(39,48)(40,42);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20)(18,24,26,31)(19,28)(21,34)(22,33)(23,36,27,41)(25,39,40,38)(29,45)(30,44)(32,35)(37,48,42,47)(43,46);(17,21,22,35)(18,25,26,40)(19,29,30,46)(20,32,33,34)(23,37,27,42)(24,38,31,39)(28,43,44,45)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi

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, 97317);