#209420

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116616);