#300438

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
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	#62435	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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)(3,7)(4,13)(5,9)(6,10)(8,14)(11,16)(12,15)(17,18,24,20)(19,27,31,28)(21,23,26,22)(25,30,32,29);(1,3)(2,7)(4,14)(5,11)(6,12)(8,13)(9,16)(10,15)(17,19,22,29,24,31,23,30)(18,25,26,27,20,32,21,28);(1,4,5,3,6,11,14,12)(2,7,8,13,10,15,9,16)(17,20,22,21,24,18,23,26)(19,27,30,25,31,28,29,32);(17,21,22,18,24,26,23,20)(19,28,29,25,31,27,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,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,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 207634);