#376268

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250083, #72256865, #72256881, …
maximal normal subgroup of	#72250083, #72256865, #72256881, …
maximal quotient of	#72308389, #72343040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283464);