#188171

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2210 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 2210 conjugacy classes of subgroups — too many to draw. Only its 310 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,27,21,26)(19,29,32,38,33,24,31,39)(20,25,34,40,35,41,36,42)(30,43,44,48,45,37,46,47);(4,12)(7,14)(10,16)(13,15)(17,19,23,33)(18,24,27,29)(20,30,35,45)(21,32,22,31)(25,37,41,43)(26,39,28,38)(34,46,36,44)(40,48,42,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,41)(28,40)(29,43)(31,46)(32,44)(33,45)(38,48)(39,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,34,35,36)(24,38,29,39)(25,40,41,42)(30,44,45,46)(37,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95367);