#232492

order	512
nilpotency class	3
isoclinism	isoclinic class of #232487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1395 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 196 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) = 128

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

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

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, 139688);