#7614469

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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)(14,19)(15,23)(16,17)(20,27)(21,22)(24,25);(2,4)(6,10)(11,12)(14,19)(15,24)(16,17)(21,22)(23,25)(26,28);(5,7,9,8)(6,11,10,12)(13,14,16,19,18,21,17,22)(15,20,23,26,25,27,24,28);(1,2,3,4)(13,15,16,23,18,25,17,24)(14,20,19,26,21,27,22,28);(5,8,9,7)(6,11,10,12)(14,21)(19,22)(20,27)(26,28)

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,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	#72257074, #72257119, #72257166
maximal normal subgroup of	#72257074, #72257119, #72257166
maximal quotient of	#72265653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521665);