#6584039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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
prime radical	#24384	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2262 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)(7,12)(8,10)(9,11)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,19)(15,18,22,25)(16,20,23,26)(21,24,27,28);(2,4)(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(14,19)(18,25)(20,26)(24,28);(2,4)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257442, #72257534, #72263855
maximal normal subgroup of	#72257442, #72257534, #72263855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491235);