#7941722

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	#37667	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4691 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,29)(21,30)(22,25)(23,24)(27,28)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,25)(18,22,24,26)(20,27,30,31)(21,28,29,32);(2,6)(5,7)(9,11,13,12)(10,14,16,15)(18,24)(19,25)(20,30)(28,32);(1,2)(3,5)(4,6)(7,8)(17,20)(18,21)(19,27)(22,28)(23,30)(24,29)(25,31)(26,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,21)(18,20)(19,28)(22,27)(23,29)(24,30)(25,32)(26,31);(17,22)(18,25)(19,24)(20,28)(21,31)(23,26)(27,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848918);