#511852

order	512
nilpotency class	4
isoclinism	isoclinic class of #511849
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261390
maximal normal subgroup of	#72261390

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419048);