#210319

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,22,25,23,26,21,19)(20,24,30,28,31,32,29,27);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19)(18,22)(20,28)(21,26)(23,25)(24,29)(27,31)(30,32);(1,3,6,12)(2,8,10,16)(4,13,5,11)(7,14,9,15)(17,18,23,26)(19,22,25,21)(20,24,31,32)(27,30,28,29);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(17,20,21,29,23,31,22,30)(18,24,19,27,26,32,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117515);