#177750

order	512
nilpotency class	3
isoclinism	isoclinic class of #177733
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	#2679	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

962 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

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

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

Polycyclic presentation:

2^9|a^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,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84946);