#395255

order	512
nilpotency class	3
isoclinism	isoclinic class of #395254
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1748 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)(3,7)(4,15)(5,13)(6,14)(8,11)(9,18)(10,20)(12,19)(16,29)(17,24)(21,26)(22,25)(23,32)(27,30)(28,31);(1,3,7,2,8,14,6,11)(4,9,18,15,19,24,17,12)(5,13,21,10,22,25,20,26)(16,23,31,28,32,29,30,27);(1,4)(2,9)(3,12)(5,16)(6,17)(7,18)(8,19)(10,23)(11,24)(13,27)(14,15)(20,30)(21,31)(22,32)(25,28)(26,29);(1,5,6,20,8,22,7,21)(2,10,3,13,11,26,14,25)(4,16,17,30,19,32,18,31)(9,23,12,27,24,29,15,28)

Polycyclic presentation:

2^9|b^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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	#72213913, #72215958, #72215972, …
maximal normal subgroup of	#72213913, #72215958, #72215972, …
maximal quotient of	C₂×#395255, #72255597, #72314091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302451);