#3473640

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1696 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)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,23,24,20)(16,25,26,21)(17,27,28,22)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(7,11)(10,12)(14,19)(15,24)(21,25)(22,27)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(5,7)(6,10)(8,12)(9,11)(13,15,18,24)(14,20,19,23)(16,22,26,27)(17,25,28,21)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(1,2,3,4)(13,16,18,26)(14,21,19,25)(15,22,24,27)(17,23,28,20)(29,31)(30,34)(32,41)(33,39)(35,44)(36,37)(38,42)(40,43);(5,8,9,6)(7,12,11,10)(13,17)(14,22)(15,21)(16,20)(18,28)(19,27)(23,26)(24,25)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3380836);