#7849819

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	#36152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6228 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)(11,16)(12,14)(13,15)(17,18)(19,28)(20,26)(21,30)(22,27)(23,25)(24,29)(31,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,22)(21,27)(23,29)(25,28)(26,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15)(18,25)(19,29)(20,30)(27,32);(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,28)(29,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,23,26)(18,20,25,30)(19,27,29,32)(22,28,31,24);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,27)(19,20)(21,24)(23,31)(25,32)(26,28)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262168, #72262195
maximal normal subgroup of	#72262168, #72262195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7757015);