#6457676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2085 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)(13,14,18,22,19,23,17,21)(15,24,16,27,26,25,28,20);(6,10)(11,12)(13,15)(14,20)(16,17)(18,28)(19,26)(21,24)(22,25)(23,27);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,17,15,19,28,18,26)(14,20,21,25,23,27,22,24);(5,8,9,7)(6,12,10,11)(13,14,18,22,19,23,17,21)(15,25,16,20,26,24,28,27);(13,17,19,18)(14,21,23,22)(15,16,26,28)(20,24,27,25)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=i,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364872);