#267386

order	512
nilpotency class	3
isoclinism	isoclinic class of #267371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

968 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)(3,11,14,7)(4,13)(5,12,16,9)(6,10)(8,15)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3,6,14)(2,7,10,11)(4,12,15,9)(5,13,16,8)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,4)(2,8)(3,12)(5,7)(6,15)(9,14)(10,13)(11,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,5)(2,9)(3,13)(4,7)(6,16)(8,14)(10,12)(11,15)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306412, #72306466, #72306496
maximal normal subgroup of	#72306412, #72306466, #72306496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174582);