#6598558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574008
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	#36360	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2050 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)(13,14)(15,24)(16,21)(17,22)(18,23)(19,26)(20,25)(27,28);(5,6,8,9)(7,10,11,12)(13,15,16,26,18,27,17,25)(14,19,22,24,23,20,21,28);(2,4)(6,9)(7,11)(13,15,16,26,18,27,17,25)(14,20,22,28,23,19,21,24);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,24,20,28);(6,9)(7,11)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257264, #72257414, #72263880, …
maximal normal subgroup of	#72257264, #72257414, #72263880, …
maximal quotient of	#72264289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505754);