#7682629

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	#35508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4537 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=gh,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, 7589825);