#351609

order	512
nilpotency class	3
isoclinism	isoclinic class of #347818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 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)(3,5)(4,6)(7,8)(10,15)(11,14)(12,16)(18,22)(19,26)(20,23)(24,29)(25,32)(27,30);(1,3,4,7)(2,5,6,8)(9,10,14,12,13,16,11,15)(17,18,23,29,21,24,20,22)(19,27,31,30,26,25,28,32);(2,6)(5,8)(9,11,13,14)(10,15,16,12)(18,24)(22,29)(25,27)(30,32);(9,12,14,16,13,15,11,10)(17,19,20,28,21,26,23,31)(18,25,22,30,24,27,29,32)

Polycyclic presentation:

2^9|b^2=efhi,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,g^d=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258805);