#8058977

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333432
maximal normal subgroup of	#72333432
maximal quotient of	#72333442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7966173);