#6671075

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6578271);