#254517

order	512
nilpotency class	3
isoclinism	isoclinic class of #254515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36812	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

957 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286165
maximal normal subgroup of	#72286165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161713);