#244516

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
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

1048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,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, 151712);