#536065

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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

3792 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443261);