#7520875

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4)(6,10)(11,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,25)(27,28);(2,4);(2,4)(5,7,9,8)(6,11,10,12)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(5,8,9,7)(6,11,10,12)(14,19)(18,21)(20,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307576, #72307657, #72308463
maximal normal subgroup of	#72307576, #72307657, #72308463

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428071);