#247831

order	512
nilpotency class	3
isoclinism	isoclinic class of #247783
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

1193 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,e^c=eh,f^c=fi,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, 155027);