#6466867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	#35274	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=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, 6374063);