#4381415

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

1836 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)(5,6,9,12)(7,10,8,11)(13,14)(15,21)(16,24)(17,18)(19,22)(20,26)(23,27)(25,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,8)(6,10,12,11)(14,18)(15,22)(20,24)(23,25)(31,38)(35,37)(39,44)(42,43);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,26)(24,27)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,43,44,42);(5,8,9,7)(6,11,12,10)(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,28)(22,25)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fh,d^2=fh,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, 4288611);