#7515261

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2017 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)(7,10)(8,9)(11,12)(13,14,20,23)(15,18,22,19)(16,26,21,24)(17,27,28,25);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,17)(19,23)(20,22)(21,28)(24,25)(26,27);(5,7,8,11)(6,10,9,12)(13,16)(14,17)(15,24)(18,27)(19,25)(20,21)(22,26)(23,28);(2,4)(13,17)(14,21)(15,25)(16,23)(18,24)(19,26)(20,28)(22,27);(6,9)(10,12)(13,18,20,19)(14,22,23,15)(16,25,21,27)(17,24,28,26)

Polycyclic presentation:

2^9|a^2=i,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,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422457);