#4946164

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1838 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=f,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4853360);