#270785

order	512
nilpotency class	3
isoclinism	isoclinic class of #270747
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	#62102	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177981);