#245106

order	512
nilpotency class	3
isoclinism	isoclinic class of #245105
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	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1368 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,27,32,31);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(3,10)(6,14)(11,16)(13,15)(17,19)(18,23)(20,26)(21,22)(24,31)(25,29)(27,32)(28,30)

Polycyclic presentation:

2^9|b^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^c=eh,f^c=fi,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, 152302);