#7182161

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#30847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1866 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)(6,10)(8,11)(13,14)(15,24)(16,21)(17,19)(18,22)(20,25)(23,28)(26,27);(5,6,9,10)(7,8,12,11)(14,19)(16,20)(17,28)(21,26)(22,23)(24,27);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,20)(16,19)(17,26)(18,25)(21,28)(22,24)(23,27);(6,10)(7,12)(13,16)(14,21)(15,19)(17,24)(18,27)(20,28)(22,26)(23,25);(5,8,9,11)(6,7,10,12)(14,22)(15,25)(19,23)(21,26)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089357);