#412327

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,d^c=di

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, 319523);