#247835

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

1294 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)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(2,6)(4,13)(7,10)(8,9)(11,14)(15,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(4,13)(6,8)(10,16)(11,14)(18,24)(20,21)(23,27)(25,26)(29,30)(31,32);(1,3,5,12)(2,7,9,15)(4,14,13,11)(6,10,8,16)(18,25)(19,28)(20,21)(23,31)(24,26)(27,32)

Polycyclic presentation:

2^9|d^2=hi,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, 155031);