#244334

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	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 850 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^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, 151530);