#7680589

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	#59050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 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)(3,6)(4,8)(5,7)(10,14)(15,16)(17,18,23,20)(19,27,29,24)(21,25,31,30)(22,28,32,26);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,15,12,16)(17,19,23,29)(18,24,20,27)(21,26,31,28)(22,25,32,30);(2,7)(6,8)(9,11,13,12)(10,15,14,16)(17,18,23,20)(19,27,29,24)(21,30,31,25)(22,26,32,28);(1,2)(3,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,20,23,18)(19,27,29,24)(21,30,31,25)(22,28,32,26);(1,3,5,4)(2,6,7,8)(17,21)(18,25)(19,28)(20,30)(22,24)(23,31)(26,29)(27,32);(1,4,5,3)(2,8,7,6)(17,22,23,32)(18,26,20,28)(19,25,29,30)(21,24,31,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7587785);