#7561844

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,e^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263899, #72263940, #72264055
maximal normal subgroup of	#72263899, #72263940, #72264055
maximal quotient of	#72264952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469040);