#245114

order	512
nilpotency class	3
isoclinism	isoclinic class of #245105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36707	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1282 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152310);