#345677

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

543 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252873);