#7516117

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514060
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

2295 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:

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

Polycyclic presentation:

2^9|d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308466, #72308794
maximal normal subgroup of	#72308466, #72308794

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423313);