#146901

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,9,14,11,15,10,26)(7,16,19,30,17,32,20,35)(8,21,24,29,22,36,25,13)(12,23,28,18,27,34,31,33);(1,2,3,4)(8,22)(13,29)(16,32)(18,33)(21,36)(23,34)(24,25)(30,35);(5,7,10,19,11,17,9,20)(6,12,14,31,15,27,26,28)(8,23,25,18,22,34,24,33)(13,30,21,16,29,35,36,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fi,h^c=hi

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, 54097);