#368076

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14015
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215906, #72219684, #72219687, …
maximal normal subgroup of	#72215906, #72219684, #72219687, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275272);