#299672

order	512
nilpotency class	3
isoclinism	isoclinic class of #299653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62365	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206868);