#6457645

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#34937	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2202 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)(13,14,18,21,19,22,17,16)(15,23,24,20,26,27,25,28);(6,10)(11,12)(13,15)(14,20)(16,27)(17,24)(18,25)(19,26)(21,23)(22,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,17,22,19,21,18,14)(15,20,25,23,26,28,24,27);(5,8,9,7)(6,12,10,11)(13,16,17,22,19,21,18,14)(15,20,25,23,26,28,24,27);(13,17,19,18)(14,16,22,21)(15,24,26,25)(20,27,28,23)

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364841);