#7672216

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	#35659	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gh,e^2=i,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, 7579412);