#7671007

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	#37678	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5322 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578203);