#6453819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3271 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361015);