#209432

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×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^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, 116628);