#7631054

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538250);