#6457719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

2323 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)(13,14,17,15,18,21,22,19)(16,20,24,23,25,27,28,26);(6,10)(11,12)(14,19)(15,21)(16,24)(17,22)(23,26)(25,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,22,14,18,19,17,21)(16,23,28,20,25,26,24,27);(5,8,9,7)(6,12,10,11)(13,16,17,24,18,25,22,28)(14,20,15,23,21,27,19,26);(16,25)(20,27)(23,26)(24,28)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,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^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, 6364915);