#422817

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330013);