#507549

order	512
nilpotency class	4
isoclinism	isoclinic class of #499904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29237
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 502 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)(3,12)(4,15)(5,10)(6,21)(7,23)(8,14)(9,13)(11,20)(16,26)(17,32)(18,25)(19,30)(22,27)(24,29)(28,31);(1,3,7,6,5,14,20,19)(2,8,23,30,10,12,11,21)(4,16,24,31,17,13,18,22)(9,25,27,15,26,29,28,32);(1,4,5,17)(2,9)(3,13,14,16)(6,22,19,31)(7,24,20,18)(8,25)(10,26)(11,28)(12,29)(15,30)(21,32)(23,27);(2,10)(3,14)(6,19)(9,26)(11,23)(13,16)(22,31)(27,28)

Polycyclic presentation:

2^9|b^2=egh,c^2=g,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=fhi,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, 414745);