#178108

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

990 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228031, #72228033, #72228037
maximal normal subgroup of	#72228031, #72228033, #72228037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85304);