#7519818

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
prime radical	#30460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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)(7,11)(8,12)(9,10)(13,14)(15,23)(16,22)(17,20)(18,21)(19,25)(24,27)(26,28);(1,2)(3,4)(6,10)(11,12)(13,15,16,25)(14,19,22,23)(17,26,18,24)(20,27,21,28);(5,7,9,8)(6,11,10,12)(14,20)(15,24)(17,18)(19,23)(21,22)(25,26);(2,4)(14,21)(15,24)(17,18)(20,22)(25,26)(27,28);(5,8,9,7)(6,11,10,12)(14,22)(19,23)(20,21)(27,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257414, #72257580, #72263854, …
maximal normal subgroup of	#72257414, #72257580, #72263854, …
maximal quotient of	#72264231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427014);