#128042

order	512
nilpotency class	4
isoclinism	isoclinic class of #7313
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1304
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	C₂×#1304
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,e^d=ei,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238001, #72238002, #72238005, …
maximal normal subgroup of	#72238001, #72238002, #72238005, …
maximal quotient of	C₂×#128042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35238);