#8741971

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5793 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)(3,7)(11,12)(14,15)(17,18,21,22)(19,25,27,23)(20,24,30,28)(26,32,29,31);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,12,15)(17,18,21,22)(19,23,27,25)(20,24,30,28)(26,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(18,22)(23,25)(24,28)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265963, #72265965
maximal normal subgroup of	#72265963, #72265965
maximal quotient of	#72266573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649167);