#535278

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=fghi,e^2=fghi,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, 442474);