#5212925

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5120121);