#7564224

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257053, #72257092, #72257166
maximal normal subgroup of	#72257053, #72257092, #72257166
maximal quotient of	#72264679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471420);