#239309

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

882 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 182 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,27)(19,28,25,20)(21,26,22,29)(24,32,31,30);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,23,25)(18,24,27,31)(20,29,28,26)(21,30,22,32);(6,12)(13,16)(17,20,22,24,23,28,21,31)(18,25,26,32,27,19,29,30);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,27,29)(19,30,25,32)(20,31,28,24)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,f^c=fh,g^c=gh

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, 146505);