#244404

order	512
nilpotency class	3
isoclinism	isoclinic class of #233521
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

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

normal subgroups only

This group has 886 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)(17,18,20,24,22,25,21,26)(19,27,28,32,30,31,29,23);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,30)(18,23,25,32)(20,29,21,28)(24,31,26,27);(10,13)(11,16)(14,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(23,32)(24,26)(27,31)

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72224781, #72224787
maximal normal subgroup of	#72224781, #72224787

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151600);