#9889488

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

5250 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265954, #72265959, #72265963
maximal normal subgroup of	#72265954, #72265959, #72265963
maximal quotient of	#72266562, #72266568

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9796684);