#6457685

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2092 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,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, 6364881);