#7633198

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5006 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=i,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, 7540394);