#210275

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
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	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,22,25,23,19,21,27)(20,24,29,31,30,26,28,32);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,18,23,19)(20,26,30,24)(21,27,22,25)(28,31,29,32);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,21,28,23,30,22,29)(18,24,27,32,19,26,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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,e^b=eh,f^b=fi,d^c=dh,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, 117471);