#8509003

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

5219 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262811, #72311336, #72311396, …
maximal normal subgroup of	#72262811, #72311336, #72311396, …
maximal quotient of	#72262822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416199);