#315626

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222822);