#7169773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#59044	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 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)(6,9)(7,11)(13,14)(15,22)(16,17)(18,27)(19,21)(20,23)(24,28)(25,26);(6,9)(7,11)(13,15)(14,20)(16,23)(17,26)(18,28)(19,25)(21,24)(22,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,17,21,27)(15,23,25,28)(20,26,24,22);(6,9)(10,12)(13,17,19,27)(14,16,21,18)(15,24,25,20)(22,28,26,23);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,17,21,27)(15,23,25,28)(20,26,24,22)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7076969);