#345160

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

667 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10,13,14,12,16,11,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,5,7)(2,6,8,4)(9,11,12,13)(10,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(1,4,5,6)(2,7,8,3)(10,15)(11,13)(14,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(2,8)(3,7)(10,15)(11,13)(14,16)(19,27)(22,25)(26,28)(30,32)

Polycyclic presentation:

2^9|a^2=efh,b^2=ei,c^2=ehi,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, 252356);