#506238

order	512
nilpotency class	4
isoclinism	isoclinic class of #499904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28117
prime radical	#33530	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,11)(4,16)(5,10)(6,21)(7,23)(8,14)(9,13)(12,20)(15,26)(17,27)(18,32)(19,25)(22,30)(24,29)(28,31);(1,3,7,15,5,14,20,6)(2,8,23,21,10,11,12,26)(4,17,24,22,18,13,19,31)(9,25,28,16,27,29,30,32);(1,4)(2,9)(3,13)(5,18)(6,22)(7,24)(8,25)(10,27)(11,29)(12,30)(14,17)(15,31)(16,21)(19,20)(23,28)(26,32);(2,10)(3,14)(6,15)(9,27)(12,23)(13,17)(22,31)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413434);