#508496

order	512
nilpotency class	4
isoclinism	isoclinic class of #508496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29892
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415692);