#6343836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3517	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306367, #72306429, #72306441
maximal normal subgroup of	#72306367, #72306429, #72306441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251032);