#272568

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 179764);