#7631799

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	#36022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5124 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^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, 7538995);