#246340

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	#9938	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1210 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=h,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, 153536);