#7564640

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257400, #72257580, #72263855, …
maximal normal subgroup of	#72257400, #72257580, #72263855, …
maximal quotient of	#72264438

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471836);