#244509

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61629	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1169 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151705);