#376244

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239830, #72252977, #72256865, …
maximal normal subgroup of	#72239830, #72252977, #72256865, …
maximal quotient of	#72308389

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283440);