#249735

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 156931);