#334714

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241910);