#6502935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410131);