#535941

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×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

3202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(5,6,8,7)(10,13)(11,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(2,4)(5,7,8,6)(10,13)(14,16)(18,22)(20,29)(23,25)(26,32);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 443137);