#345703

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	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 645 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^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, 252899);