#7710475

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	#45455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5027 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=ghi,e^2=gh,f^2=gh,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, 7617671);