#209555

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#18114	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 116751);