#7811924

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	#35514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4949 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^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, 7719120);