#6459990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	#35256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2368 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)(5,6,8,9)(7,10,11,12)(13,14,18,25,20,23,19,22)(15,24,28,16,26,17,21,27);(6,9)(10,12)(13,15)(14,16)(17,22)(18,21)(19,28)(20,26)(23,27)(24,25);(1,2,3,4)(13,16,18,17,20,27,19,24)(14,15,25,28,23,26,22,21);(5,7,8,11)(6,10,9,12)(13,17,19,16,20,24,18,27)(14,21,22,26,23,28,25,15);(13,18,20,19)(14,22,23,25)(15,21,26,28)(16,17,27,24)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367186);