#7812176

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4890 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,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, 7719372);