#7519277

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,3)(9,10,14,17)(11,13,18,15)(12,19,22,24)(16,23,20,21);(1,2,4,3)(5,6)(7,8)(9,11,14,18)(10,15,17,13)(12,20,22,16)(19,23,24,21);(10,11)(13,15)(16,19)(17,18)(20,24)(21,23);(6,8)(9,12,14,22)(10,16,17,20)(11,19,18,24)(13,21,15,23);(1,3,4,2)(9,13,14,15)(10,11,17,18)(12,21,22,23)(16,19,20,24)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315301, #72315342, #72315348
maximal normal subgroup of	#72315301, #72315342, #72315348
maximal quotient of	#72315713, #72315717

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426473);