#7804833

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4800 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)(11,15,16,13)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31);(1,2)(3,7)(4,8)(5,6)(9,11,12,16)(10,13,14,15)(18,21)(19,26)(20,28)(31,32);(2,6)(4,7)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,28)(30,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29);(1,3)(2,4)(5,8)(6,7)(17,21,22,18)(19,25,26,23)(20,30,28,24)(27,32,29,31);(1,4)(2,3)(5,7)(6,8)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)

Polycyclic presentation:

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

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, 7712029);