#345967

order	512
nilpotency class	3
isoclinism	isoclinic class of #345928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,e^a=eh,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,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, 253163);