#6529716

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
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	#34936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305259, #72305276, #72305279
maximal normal subgroup of	#72305259, #72305276, #72305279
maximal quotient of	#72305904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436912);