#300108

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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,d^2=eh,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, 207304);