#244744

order	512
nilpotency class	3
isoclinism	isoclinic class of #244655
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	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151940);