#8093067

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

5533 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:

(3,8)(4,6)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,2,5,7)(3,6,8,4)(11,12)(14,16)(18,22)(23,28)(24,29)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(2,7)(3,8)(9,12,13,11)(10,14,15,16)(18,22)(19,25)(24,29)(26,31);(1,4,5,6)(2,3,7,8)(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=i,b^2=hi,c^2=i,d^2=i,e^2=hi,f^2=g,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	#72279724, #72282828, #72283006, …
maximal normal subgroup of	#72279724, #72282828, #72283006, …
maximal quotient of	#72280082, #72304076

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000263);