#4435176

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1793 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)(6,9)(7,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(30,34)(31,37)(32,39)(43,44);(15,22)(18,21)(23,28)(26,27)(29,30,33,34)(31,35,37,41)(32,36,39,42)(38,43,40,44);(5,6,8,9)(7,10,11,12)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,30)(31,35)(32,36)(33,34)(37,41)(38,43)(39,42)(40,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31)(30,35)(32,40)(33,37)(34,41)(36,44)(38,39)(42,43);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,32)(30,36)(31,38)(33,39)(34,42)(35,43)(37,40)(41,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342372);