#7785150

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	#38076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4963 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7692346);