#244348

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

718 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151544);