#7682054

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	#42997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5076 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)(3,5,7,8)(9,10,13,15)(11,12,14,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30);(1,2)(3,5)(4,6)(7,8)(9,11,13,14)(10,12,15,16)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31);(9,10)(11,12)(13,15)(14,16)(18,24)(19,21)(25,29)(27,28);(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31);(17,20)(18,25)(19,27)(21,28)(22,30)(23,31)(24,29)(26,32);(1,3,4,7)(2,5,6,8)(17,21,22,19)(18,26,24,23)(20,28,30,27)(25,32,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589250);