#8109991

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

4940 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219279, #72219285
maximal quotient of	#72219274

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017187);