#7614542

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#24256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 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)(15,22)(16,18)(17,20)(19,23)(21,26)(24,25)(27,28);(2,4)(6,10)(11,12)(13,15,17,23)(14,19,20,22)(16,25,21,28)(18,27,26,24);(5,7,9,8)(6,11,10,12)(13,16,26,20,17,21,18,14)(15,19,27,28,23,22,24,25);(1,2,3,4)(13,16,26,20,17,21,18,14)(15,19,27,28,23,22,24,25);(5,8,9,7)(6,11,10,12)(14,20)(16,21)(19,22)(25,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=hi,e^2=fi,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,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257113, #72257139, #72257181
maximal normal subgroup of	#72257113, #72257139, #72257181
maximal quotient of	#72308475

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521738);