#7515933

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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)(7,12)(8,10)(9,11)(13,14)(15,17)(16,22)(18,23)(19,20)(21,27)(24,28)(25,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,24)(14,17,20,28)(16,21,18,25)(22,27,23,26);(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,24)(17,27)(18,19)(26,28);(2,4)(13,17)(14,21)(15,22)(16,26)(18,27)(19,28)(20,25)(23,24);(6,11)(10,12)(13,16,19,18)(14,22,20,23)(15,25,24,21)(17,26,28,27)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,e^2=i,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,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257339, #72257588, #72257752
maximal normal subgroup of	#72257339, #72257588, #72257752
maximal quotient of	#72264575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423129);