#177797

order	512
nilpotency class	3
isoclinism	isoclinic class of #177733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1125 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1125 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(5,8)(10,15)(11,16)(12,13)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,16,13,15)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26);(2,6)(5,8)(9,12,14,13)(10,15,11,16)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,26,32,31)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,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,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72217689, #72217722, #72217728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84993);