#6596550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3178 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)(13,14,17,21,19,23,18,24)(15,22,25,28,16,27,26,20);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,26)(18,25)(21,27)(22,24)(23,28);(6,9)(7,11)(14,21)(15,16)(17,18)(20,22)(23,24)(27,28);(5,7)(6,10)(8,11)(9,12)(13,16,19,15)(14,22,23,27)(17,26,18,25)(20,21,28,24);(6,9)(7,11)(14,23)(15,16)(21,24)(25,26)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263923, #72263976, #72264047
maximal normal subgroup of	#72263923, #72263976, #72264047
maximal quotient of	#72264320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503746);