#6441767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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
prime radical	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2245 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)(7,9)(8,10)(11,12)(13,14,17,15,19,22,18,21)(16,24,25,28,27,20,26,23);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,20)(18,22)(19,21)(23,26)(24,27)(25,28);(2,4)(6,10)(9,12)(13,14,17,15,19,22,18,21)(16,20,25,23,27,24,26,28);(2,4)(13,16,17,25,19,27,18,26)(14,20,15,23,22,24,21,28);(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,28,24,23)

Polycyclic presentation:

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

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, 6348963);