#7631821

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	#36225	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539017);