#177930

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 704 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(14,21)(22,23)(25,26,30,34)(27,35,37,48)(28,32,39,46)(29,41,43,33)(31,42,45,38)(36,49,51,56)(40,53,52,47)(44,54,55,50);(9,10,13,16)(11,14,18,21)(12,19,20,15)(17,23,24,22)(25,27)(26,31)(28,36)(29,42)(30,37)(32,44)(33,48)(34,45)(35,41)(38,43)(39,51)(40,54)(46,55)(47,56)(49,53)(50,52);(2,5)(3,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,23)(20,24)(25,28,30,39)(26,32,34,46)(27,36,37,51)(29,40,43,52)(31,44,45,55)(33,47,41,53)(35,49,48,56)(38,50,42,54);(2,6)(5,7)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,22,21,23)(27,37)(28,39)(31,45)(32,46)(35,48)(38,42)(40,52)(47,53)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85126);