#8915206

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

6128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249434, #72249510, #72249517
maximal normal subgroup of	#72249434, #72249510, #72249517
maximal quotient of	#72262673, #72262822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8822402);