#270703

order	512
nilpotency class	3
isoclinism	isoclinic class of #270683
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

859 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:

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

Polycyclic presentation:

2^9|a^2=f,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,e^b=eh,g^b=gi,d^c=di,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, 177899);