#6441801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2344 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,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^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, 6348997);