#577216

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2846 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)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,19,20,17)(22,26)(23,27);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,15,16,20)(14,17,18,19)(21,22)(23,24)(25,26)(27,28);(1,3,4,2)(7,11)(9,12)(14,18)(17,19)(22,26)(24,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,23)(22,24)(25,27)(26,28);(5,7)(6,9)(8,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=fhi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 484412);