#6457678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35528	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2293 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6364874);