#515259

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

4690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 422455);