#394806

order	512
nilpotency class	3
isoclinism	isoclinic class of #394805
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14719
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1592 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)(3,6)(4,15)(5,10)(7,11)(8,12)(9,18)(13,19)(14,20)(16,23)(17,27)(21,25)(22,26)(24,31)(28,32)(29,30);(1,3,7,12,8,11,6,2)(4,16,23,15,19,9,18,13)(5,14,21,26,22,25,20,10)(17,29,31,28,32,24,30,27);(1,4,6,18,8,19,7,23)(2,9,11,15,12,16,3,13)(5,17,20,30,22,32,21,31)(10,24,25,28,26,29,14,27);(1,5)(2,10)(3,14)(4,17)(6,20)(7,21)(8,22)(9,24)(11,25)(12,26)(13,27)(15,28)(16,29)(18,30)(19,32)(23,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316368, C₂×#394806, #72338487, …
maximal normal subgroup of	#72316368, C₂×#394806, #72338487, …
maximal quotient of	C₂×#394806, #72316819, #72316924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302002);