#7684831

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	#43029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4759 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)(4,8)(5,6)(10,13)(14,16)(17,18)(19,21)(20,26)(22,27)(23,28)(24,30)(25,31)(29,32);(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,22)(21,28)(23,30)(25,27)(26,32)(29,31);(2,6)(3,8)(17,20,23,31)(18,25,28,26)(19,29,30,22)(21,27,24,32);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,22)(21,27)(23,31)(24,32)(25,28)(29,30);(1,4,5,7)(2,8,6,3)(17,21,23,24)(18,19,28,30)(20,27,31,32)(22,25,29,26);(17,22)(18,27)(19,20)(21,26)(23,29)(24,25)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=h,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, 7592027);