#8093288

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

5028 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18,22,25)(19,26,29,23)(20,24,30,21)(27,28,32,31);(2,6)(3,8)(11,16)(12,15)(20,30)(21,24)(27,32)(28,31);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,28)(29,32);(1,2)(3,7)(4,8)(5,6)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,26,32);(1,4,5,7)(2,8,6,3)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303876
maximal normal subgroup of	#72303876
maximal quotient of	#72304074, #72304090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000484);