#7712329

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(18,23)(19,24)(25,29)(26,31);(2,7)(4,6)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(1,4,5,6)(2,3,7,8);(1,3)(2,6)(4,7)(5,8)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,27,28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619525);