#345328

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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, 252524);