#7985152

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=gh,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, 7892348);