#10328251

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

5223 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10235447);