#235203

order	512
nilpotency class	3
isoclinism	isoclinic class of #235187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60958	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1321 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,16)(14,15)(18,22)(19,25)(20,28)(24,29)(27,30)(31,32);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,18,21,24)(19,26,27,23)(20,29,28,22)(25,32,30,31);(1,3)(2,8)(4,13)(5,12)(6,14)(7,15)(9,11)(10,16)(17,19)(18,23)(20,30)(21,27)(22,31)(24,26)(25,28)(29,32);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,11,16,15)(19,27)(23,26)(25,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263908, #72263912, #72263924
maximal normal subgroup of	#72263908, #72263912, #72263924
maximal quotient of	#72264295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142399);