#8150137

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

5509 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261968, #72261985, #72262162
maximal normal subgroup of	#72261968, #72261985, #72262162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8057333);