#270825

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	#3480	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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,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, 178021);