#300097

order	512
nilpotency class	3
isoclinism	isoclinic class of #300077
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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 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)(3,11)(4,14)(5,16)(6,8)(7,13)(9,15)(10,12)(17,18,24,19)(20,21,32,31)(22,28,23,27)(25,26,29,30);(1,3,5,13,6,4,12,15)(2,7,10,11,8,9,16,14)(17,19,22,28,24,18,23,27)(20,31,26,29,32,21,30,25);(2,8)(3,4)(10,16)(13,15)(17,20,22,26,24,32,23,30)(18,25,27,31,19,29,28,21);(1,4,5,15,6,3,12,13)(2,9,10,14,8,7,16,11)(17,21,22,25,24,31,23,29)(18,26,27,32,19,30,28,20)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207293);