#303275

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39415	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1065 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)(3,7)(4,6)(5,8)(9,10,15,11)(12,14,16,13)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,13,16,14)(17,19)(18,23)(20,31)(21,27)(22,28)(24,26)(25,32)(29,30);(1,3,5,4)(2,7,8,6)(9,12,14,11,15,16,13,10)(17,20,30,18,21,25,22,24)(19,23,29,32,27,26,28,31);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,12,11,16)(18,24)(20,25)(23,26)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210471);