#272109

order	512
nilpotency class	3
isoclinism	isoclinic class of #272091
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=di,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, 179305);