#6440777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2886 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)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,26,16)(17,22,18,28)(20,21,27,25);(1,2)(3,4)(6,9)(10,12)(13,15,18,25,19,26,17,21)(14,20,22,24,23,27,28,16);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,20,19,24,17,27)(14,21,22,15,23,25,28,26);(2,4)(13,17,19,18)(14,22,23,28)(15,25,26,21)(16,27,24,20);(13,17,19,18)(14,22,23,28)(15,21,26,25)(16,20,24,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=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, 6347973);