#8093071

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5892 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304009
maximal normal subgroup of	#72304009
maximal quotient of	#72303956

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000267);