#7003306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42916	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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)(7,9)(8,10)(11,12)(14,20)(15,25)(17,18)(21,26)(22,28)(23,24);(5,6)(7,11)(8,10)(9,12)(13,14,18,20,19,24,17,23)(15,22,16,26,25,21,27,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,18)(17,27)(19,25)(20,28)(22,24)(23,26);(7,12)(9,11)(13,16,19,27)(14,22,24,21)(15,17,25,18)(20,28,23,26);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,16,25,27)(21,28,22,26)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910502);