#4417672

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(7,11)(9,12)(13,14,18,19)(15,20,22,16)(17,21,26,27)(23,28,25,24)(30,31)(32,39)(33,41)(42,43);(5,6,8,10)(7,9,11,12)(14,19)(15,22)(21,27)(23,25)(29,30,34,31)(32,35,39,37)(33,36,41,38)(40,42,44,43);(1,2,3,4)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,26)(27,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(5,7,8,11)(6,9,10,12)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,28)(22,25)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=fi,e^2=fh,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, 4324868);