#6583923

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

2207 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=fi,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, #72263976
maximal normal subgroup of	#72263949, #72263976
maximal quotient of	#72264161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491119);