#7780299

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	#45439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7687495);