#8756116

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

5220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319702, #72319825
maximal normal subgroup of	#72319702, #72319825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663312);