#257957

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165153);