#6575552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
prime radical	#35821	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2227 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)(7,10)(8,9)(11,12)(13,14,18,16)(15,23,26,20)(17,21,25,27)(19,22,24,28);(6,9)(10,12)(13,15,18,26)(14,20,16,23)(17,24,25,19)(21,22,27,28);(2,4)(13,16,18,14)(15,21,26,27)(17,23,25,20)(19,22,24,28);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,24)(16,27)(18,25)(19,26)(20,22)(23,28);(15,25)(17,26)(19,24)(20,21)(22,28)(23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482748);