#5477844

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1700 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)(6,7)(8,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,32)(33,38)(34,35)(36,37)(39,40)(41,44)(42,43);(5,6,9,7)(8,10,12,11)(15,22)(17,25)(19,21)(20,24)(30,35)(31,37)(38,43)(39,41);(5,7,9,6)(8,10,12,11)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,36)(32,35)(33,41)(34,37)(38,40)(39,42)(43,44);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,36)(30,37,35,31)(33,40,42,44)(38,41,43,39);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33)(30,38)(31,39)(32,40)(34,42)(35,43)(36,44)(37,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,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, 5385040);