#7170297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 7077493);