#531990

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4)(5,6,9,10)(7,11,8,12)(14,17)(15,19)(22,24)(23,27);(6,10)(11,12)(13,14,16,17)(15,18,19,20)(21,22,25,24)(23,28,27,26);(5,7,9,8)(6,11,10,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,28)(25,27);(5,8,9,7)(6,12,10,11)(13,15)(14,18)(16,19)(17,20)(21,24,25,22)(23,28,27,26);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 439186);