#232862

order	512
nilpotency class	3
isoclinism	isoclinic class of #232862
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1386 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338241, #72338244, #72338245
maximal normal subgroup of	#72338241, #72338244, #72338245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140058);