#354424

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1826 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)(5,6,8,7)(13,14,17,19,18,22,16,21)(15,20,25,26,23,27,24,28);(9,10,11,12)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(2,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(5,7,8,6)(15,23)(20,27)(24,25)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi,d^c=di

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, 261620);