#4414409

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4321605);