#272055

order	512
nilpotency class	3
isoclinism	isoclinic class of #272027
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	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 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:

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

Polycyclic presentation:

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

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, 179251);