#7631837

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4858 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,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, 7539033);