#6491312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
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×#3482	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3114 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:

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

Polycyclic presentation:

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

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, 6398508);