#261029

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,30)(28,32);(2,5)(6,7)(9,10,12,16)(11,15,13,14)(17,19,21,20)(18,23,25,27)(22,29,30,28)(24,31,32,26);(10,14)(11,13)(15,16)(17,20,21,19)(18,24,25,32)(22,28,30,29)(23,26,27,31);(1,3,4,8)(2,6,5,7)(9,11,12,13)(10,15,16,14)(18,25)(23,27)(24,32)(26,31)

Polycyclic presentation:

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

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, 168225);