#6542464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3365 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)(15,25)(16,27)(17,23)(18,22)(19,24)(20,26)(21,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,24,21)(17,26,18,28)(22,25,23,27);(6,9)(10,12)(13,16,19,15)(14,21,24,20)(17,28,18,26)(22,27,23,25);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,26,16,28)(20,25,21,27);(7,11)(10,12)(13,17,19,18)(14,23,24,22)(15,26,16,28)(20,27,21,25)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,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	#72284414, #72284714, #72285105
maximal normal subgroup of	#72284414, #72284714, #72285105

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449660);