#7517302

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2043 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257505, #72257569, #72263868, …
maximal normal subgroup of	#72257505, #72257569, #72263868, …
maximal quotient of	#72307889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424498);