#199135

order	512
nilpotency class	3
isoclinism	isoclinic class of #198834
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1487 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257620, #72257646, #72257669
maximal normal subgroup of	#72257620, #72257646, #72257669
maximal quotient of	#72240844, #72264251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106331);