#7182711

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22)(23,30)(24,32)(25,26)(27,29)(28,34)(31,35)(33,36);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,30,22,27,25,26,29)(24,31,36,28,34,33,35,32);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,34)(22,28,29,32)(23,31,25,33)(26,35,30,36);(7,14)(10,13)(15,20)(18,19)(21,25,27,23)(22,26,29,30)(24,33,34,31)(28,35,32,36);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(24,34)(28,32)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7089907);