#6441786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

2256 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)(13,14,17,20,18,21,25,15)(16,19,23,27,24,26,28,22);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,15)(16,23)(17,25)(20,21)(22,27)(24,28);(2,4)(6,10)(9,12)(13,15,17,14,18,20,25,21)(16,22,23,19,24,27,28,26);(2,4)(13,16,17,23,18,24,25,28)(14,19,20,27,21,26,15,22);(16,24)(19,26)(22,27)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6348982);