#7515135

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

2101 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,26,16)(17,23,18,21)(22,28,25,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,25)(18,22)(19,26)(21,28)(23,27);(5,7,8,11)(6,10,9,12)(14,21)(15,22)(17,18)(23,24)(25,26)(27,28);(2,4)(13,16,19,20)(14,22,24,25)(15,23,26,21)(17,27,18,28);(6,9)(10,12)(13,17,19,18)(14,23,24,21)(15,22,26,25)(16,27,20,28)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,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,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422331);