#246351

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
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
prime radical	#46428	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1299 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)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(6,15)(10,16)(17,20,21,27)(18,23,24,30)(19,25,26,32)(22,28,29,31);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)

Polycyclic presentation:

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

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, 153547);