#7631829

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5282 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)(17,18,23,27)(19,24,29,32)(20,30,28,25)(21,26,31,22);(9,10,14,11)(12,15,13,16)(17,19)(18,24)(20,31)(21,28)(22,30)(23,29)(25,26)(27,32);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,16,13,15)(17,20,23,28)(18,25,27,30)(19,21,29,31)(22,32,26,24);(9,12,14,13)(10,15,11,16)(17,19)(18,24)(20,21)(22,25)(23,29)(26,30)(27,32)(28,31);(17,21)(18,26)(19,28)(20,29)(22,27)(23,31)(24,25)(30,32);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,11,15)(17,22,23,26)(18,21,27,31)(19,25,29,30)(20,32,28,24)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,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, 7539025);