#345978

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ei,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 253174);