#7614508

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(15,22)(16,24)(17,26)(18,21)(19,27)(20,23)(25,28);(2,4)(6,10)(11,12)(13,15,18,23)(14,20,21,22)(16,25,27,26)(17,19,28,24);(5,7,9,8)(6,11,10,12)(13,16,24,14,18,27,19,21)(15,22,28,26,23,20,17,25);(1,2,3,4)(13,17,19,23,18,28,24,15)(14,20,16,26,21,22,27,25);(5,8,9,7)(6,11,10,12)(14,21)(16,27)(20,22)(25,26)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257096, #72257139, #72257166, …
maximal normal subgroup of	#72257096, #72257139, #72257166, …
maximal quotient of	#72264183, #72264192, #72308907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521704);