#6583917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2311 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,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.

Relationships

maximal subgroup of	#72263949, #72263962
maximal normal subgroup of	#72263949, #72263962
maximal quotient of	#72265034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491113);