#4435355

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1938 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)(7,11,12,9)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(31,37)(33,41)(35,38)(36,40);(14,18)(19,21)(20,24)(27,28)(29,30,34,32)(31,38,37,35)(33,36,41,40)(39,44,43,42);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342551);