#245205

order	512
nilpotency class	3
isoclinism	isoclinic class of #245204
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1288 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1288 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152401);