#7631832

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

5408 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)(17,18,24,26)(19,20,22,21)(23,25,31,32)(27,28,30,29);(9,10,14,11)(12,15,13,16)(17,19,24,22)(18,21,26,20)(23,30,31,27)(25,28,32,29);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,16,13,15)(17,20,24,21)(18,19,26,22)(23,28,31,29)(25,27,32,30);(9,12,14,13)(10,15,11,16)(17,21,24,20)(18,19,26,22)(23,29,31,28)(25,27,32,30);(9,12,14,13)(10,15,11,16)(17,22,24,19)(18,21,26,20)(23,30,31,27)(25,29,32,28);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,11,15)(17,23,24,31)(18,25,26,32)(19,27,22,30)(20,28,21,29)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,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, 7539028);