#4389600

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1745 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)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,16,23,20)(17,21,26,27)(22,24,25,28)(30,34)(32,40)(35,37)(38,41);(14,19)(16,20)(21,27)(24,28)(29,30,33,34)(31,37,39,35)(32,36,40,42)(38,44,41,43);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,19)(17,25)(18,23)(21,24)(22,26)(27,28)(29,31,33,39)(30,35,34,37)(32,41,40,38)(36,44,42,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,15,19,23)(17,24,26,28)(21,22,27,25)(29,30)(31,35)(32,36)(33,34)(37,39)(38,43)(40,42)(41,44);(5,7,8,11)(6,9,10,12)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,28)(23,25)(29,32,33,40)(30,36,34,42)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 4296796);