#7684665

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591861);