#10521345

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

5303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311201, #72318542
maximal normal subgroup of	#72311201, #72318542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428541);