#331477

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238673);