#270680

order	512
nilpotency class	3
isoclinism	isoclinic class of #270619
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×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177876);