#6470875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470874
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	#35077	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,23)(15,24,25,28)(16,22,17,21)(19,27,20,26);(5,7)(6,9)(8,11)(10,12)(13,15,16,26,18,25,17,27)(14,19,21,24,23,20,22,28);(2,4)(7,11)(9,12)(13,15,16,26,18,25,17,27)(14,20,21,28,23,19,22,24);(7,11)(9,12);(6,10)(9,12)(15,25)(19,20)(24,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378071);