#395193

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14722
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1768 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)(3,14)(4,19)(5,13)(6,11)(7,12)(8,16)(9,21)(10,17)(15,25)(18,24)(20,27)(22,26)(23,30)(28,29)(31,32);(1,3,5,16,7,18,6,17)(2,8,11,14,12,10,13,24)(4,15,20,29,22,30,21,31)(9,25,26,32,27,23,19,28);(1,4)(2,9)(3,15)(5,20)(6,21)(7,22)(8,23)(10,25)(11,26)(12,27)(13,19)(14,28)(16,29)(17,31)(18,30)(24,32);(1,3,5,16,7,18,6,17)(2,10,11,24,12,8,13,14)(4,15,20,29,22,30,21,31)(9,25,26,32,27,23,19,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215972, #72216071, #72216123, …
maximal normal subgroup of	#72215972, #72216071, #72216123, …
maximal quotient of	#72239647, #72239648, C₂×#395193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302389);