#7985083

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892279);