#270863

order	512
nilpotency class	3
isoclinism	isoclinic class of #270811
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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^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,d^b=dh,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=gi,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, 178059);