#7811999

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	#58991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5275 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:

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

Polycyclic presentation:

2^9|b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7719195);