#8455753

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

5949 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)(4,6)(5,8)(10,14)(11,16)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25);(9,11)(10,15)(12,14)(13,16)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32);(2,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32);(1,3)(2,5)(4,8)(6,7)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,14)(13,15)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256737, #72263362
maximal normal subgroup of	#72256737, #72263362
maximal quotient of	#72256786, #72263408

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362949);